StudySmarter: Study help & AI tools

4.5 • +22k Ratings

More than 22 Million Downloads

Free

Area under curve

Dive into the fascinating subject of 'Area under Curve' in engineering mathematics. This thorough exploration sheds light on fundamental concepts, practical applications, and varied methods employed in curve calculations. Topics covered include everything from basic expeditions into the Area under Curve calculus to more in-depth discussions on its integration and formula. Practical examples and comparisons of exact and approximate methods provide further understanding. Whether a seasoned scholar or a novice in engineering mathematics, this study will enhance your comprehension of the 'Area under Curve'.

Explore our app and discover over 50 million learning materials for free.

- Design Engineering
- Engineering Fluid Mechanics
- Engineering Mathematics
- Acceptance Sampling
- Addition Rule of Probability
- Algebra Engineering
- Application of Calculus in Engineering
- Area under curve
- Basic Algebra
- Basic Derivatives
- Basic Matrix Operations
- Bayes' Theorem
- Binomial Series
- Bisection Method
- Boolean Algebra
- Boundary Value Problem
- CUSUM
- Cartesian Form
- Causal Function
- Centroids
- Cholesky Decomposition
- Circular Functions
- Complex Form of Fourier Series
- Complex Hyperbolic Functions
- Complex Logarithm
- Complex Trigonometric Functions
- Conservative Vector Field
- Continuous and Discrete Random Variables
- Control Chart
- Convergence Engineering
- Convergence of Fourier Series
- Convolution Theorem
- Correlation and Regression
- Covariance and Correlation
- Cramer's rule
- Cross Correlation Theorem
- Curl of a Vector Field
- Curve Sketching
- D'alembert Wave Equation
- Damping
- Derivative of Polynomial
- Derivative of Rational Function
- Derivative of a Vector
- Directional Derivative
- Discrete Fourier Transform
- Divergence Theorem
- Divergence Vector Calculus
- Double Integrals
- Eigenvalue
- Eigenvector
- Engineering Analysis
- Engineering Graphs
- Engineering Statistics
- Euler's Formula
- Exact Differential Equation
- Exponential and Logarithmic Functions
- Fourier Coefficients
- Fourier Integration
- Fourier Series
- Fourier Series Odd and Even
- Fourier Series Symmetry
- Fourier Transform Properties
- Fourier Transform Table
- Gamma Distribution
- Gaussian Elimination
- Half Range Fourier Series
- Higher Order Integration
- Hypergeometric Distribution
- Hypothesis Test for a Population Mean
- Implicit Function
- Improved Euler Method
- Interpolation
- Inverse Laplace Transform
- Inverse Matrix Method
- Inverse Z Transform
- Jacobian Matrix
- Laplace Shifting Theorem
- Laplace Transforms
- Large Sample Confidence Interval
- Least Squares Fitting
- Logic Gates
- Logical Equivalence
- Maths Identities
- Maxima and Minima of functions of two variables
- Maximum Likelihood Estimation
- Mean Value and Standard Deviation
- Method of Moments
- Modelling waves
- Multiple Regression
- Multiple Regression Analysis
- Newton Raphson Method
- Non Parametric Statistics
- Nonlinear Differential Equation
- Nonlinear Regression
- Numerical Differentiation
- Numerical Root Finding
- One Way ANOVA
- P Value
- Parseval's Theorem
- Partial Derivative
- Partial Derivative of Vector
- Partial Differential Equations
- Particular Solution for Differential Equation
- Phasor
- Piecewise Function
- Polar Form
- Polynomial Regression
- Probability Engineering
- Probability Tree
- Quality Control
- RMS Value
- Radians vs Degrees
- Rank Nullity Theorem
- Rank of a Matrix
- Reliability Engineering
- Runge Kutta Method
- Scalar & Vector Geometry
- Second Order Nonlinear Differential Equation
- Simple Linear Regression Model
- Single Sample T Test
- Standard Deviation of Random Variable
- Superposition
- System of Differential Equations
- System of Linear Equations Matrix
- Taylor's Theorem
- Three Way ANOVA
- Total Derivative
- Transform Variables in Regression
- Transmission Line Equation
- Triple Integrals
- Triple Product
- Two Sample Test
- Two Way ANOVA
- Unit Vector
- Vector Calculus
- Wilcoxon Rank Sum Test
- Z Test
- Z Transform
- Z Transform vs Laplace Transform
- Engineering Thermodynamics
- Materials Engineering
- Professional Engineering
- Solid Mechanics
- What is Engineering

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmeldenDive into the fascinating subject of 'Area under Curve' in engineering mathematics. This thorough exploration sheds light on fundamental concepts, practical applications, and varied methods employed in curve calculations. Topics covered include everything from basic expeditions into the Area under Curve calculus to more in-depth discussions on its integration and formula. Practical examples and comparisons of exact and approximate methods provide further understanding. Whether a seasoned scholar or a novice in engineering mathematics, this study will enhance your comprehension of the 'Area under Curve'.

For example, suppose you want to find the area under a curve \(y = x^2\) from \(x = 0\) to \(x = 2\). You would find this by evaluating: \[ A = \int_{0}^{2} x^2 \, dx \]

The area under the curve equals the sum of the areas of all the rectangles. Mathematically, this is represented as the limit of a sum (integral) as the width of the rectangles approaches zero.

The deep dive: both definite and indefinite integrals are not just abstract mathematical operations, but are fundamental to many practical applications in physics and engineering. This includes solving problems in fields like electrostatics, hydrodynamics, and signal processing.

- Physics: It can be used to calculate the distance travelled by an object when you know the object's speed at different moments in time.
- Engineering: In electrical engineering, the area under the power-time curve gives the amount of work done or energy used.
- Economics: It is used in determining the total cost of production over a certain period, given the rate of production.

- \(\Delta x = \frac{b - a}{n}\) is the width of each subinterval,
- \(x_i^*\) is a sample point in the \(i\)-th subinterval.

function riemannSum(func, a, b, n) { let sum = 0; let delta = (b - a) / n; for(let i = 0; i < n; i++) { let xi = a + i * delta; sum += func(xi) * delta; } return sum; }This is a simple JavaScript function named riemannSum which calculates the Riemann sum for a given function func over the interval [a, b], and using a specified number of subintervals n. Although a Riemann sum is only an approximation—since the use of rectangles inherently disregards area that lies above the function but within the curve—it nonetheless acts as a sturdy foundation upon which integral calculus is built.

- Divide the area under the curve into an even number of intervals of equal width.
- Substitute these values into the trapezoidal rule formula.

- \(\Delta x = \frac{b - a}{n}\) is the width of each subinterval,
- \(f()\), \(a\), and \(b\) are the same functions and limits as in the previous concept.

- \(\Delta x = \frac{b - a}{n}\) is the width of each subinterval,
- \(f()\), \(a\), and \(b\) are identical to their representations in prior topics,
- \(n\) is an even number.

Now, let's see an example of how you might use the area under the curve formula for calculation. Suppose we aim to find the area under the curve for the basic function \(f(x) = x^2\) for \(x\) in the range of 0 to 2. According to our integral formula, we would do this by evaluating: \[ A = \int_{0}^{2} x^2 \, dx \] Here, \(x^2\) is the function, and 0 and 2 are the limits. The integral symbol \( \int \) is followed by the limits lower and upper of integration. \( dx \) instructs us to integrate with respect to \( x \).

**General applicability**: Integration is a general-purpose tool that can be used to find the area under curves of any shape. It's not restricted to certain classes of functions or specific curve types.**Accuracy**: Because integration works by dividing the area into infinitely many infinitesimally small rectangles, it can provide a very accurate calculation of the area under the curve, especially for smooth functions.**Extendibility**: Besides finding areas, integration is also extendable to numerous other applications. It's used for solving differential equations, finding volumes of solids of revolution, calculating work done in physics, and more.

For a simple example, say you're trying to estimate the area under the graph of the function \(f(x) = x^2\) between \(x = 1\) and \(x = 4\). You might divide this region into three rectangles of equal width. Their heights would be determined by the value of the function at their right hand side. Hence, your estimate of the area using Riemann Sums would be \[ f(2) \times 1 + f(3) \times 1 + f(4) \times 1 = 4 + 9 + 16 = 29 \] Though it's only an approximation, it helps get a basic sense of the area under the curve.

**Complex Functions:**Some functions can be complex and difficult to deal with when it comes to calculating the exact area under the curve. Approximation methods can provide a simpler way to get a good estimate of the area.**Real-time Applications:**In real-time applications, it might not always be possible to perform detailed calculations. Approximate methods are faster and can provide quick estimates.**Lack of Analytical Solution:**Not all functions have an analytical integral. In such cases, numerical approximation methods are the only way to estimate the area under the curve.

Method | Pros | Cons |

Exact | Integrals provide the exact area under the function curve between two points. This can be invaluable for functions that are simple and have a known primitive function. | The process of finding the primitive function and computing the definite integral for complex functions can be tedious and difficult. Some functions cannot be integrated using elementary functions. |

Approximate | Approximate methods provide a way to estimate difficult integrals or perform quick calculations. They can also handle functions that do not have an elementary integral. | The accuracy of an approximation can be influenced by the choice of method and the number of partitions. Better approximations usually require more calculations which can increase the computational time. |

**Contextual Understanding:**Simple functions are one thing, but more complex functions that describe physical, biological or economic processes require a contextual understanding to truly grasp their implications and applications.**Visualisation:**When you can visualise what a function represents, whether it's the trajectory of a projectile, the distribution of a population or the kinetic energy of an object, it becomes easier to understand the significance of the area under the curve.**Appreciation of Mathematics:**Seeing how these abstract mathematical concepts apply to real-life scenarios encourages greater appreciation for the field.

**Area under Curve Riemann Sum Meaning:**A method used to approximate the area under a curve, developed by the German mathematician Bernhard Riemann. It approximates the area using a set of rectangles.**Trapezoidal Rule:**An alternative method to Riemann sum for calculating area under the curve. This method uses trapezoids instead of rectangles to approximate the area under a curve, dealing more effectively with slopes.**Simpson's Rule:**A technique that provides a higher accuracy estimate for the area under a curve using parabolic arcs, greatly improving the approximation especially for curves requiring higher precision.**Area under Curve Integration:**A mathematical means to accurately calculate the area under a curve. The integral approach involves subdividing the area into an infinite count of infinitesimally narrow rectangles and summing their areas.**Area under Curve Approximation:**A method used when exact approach is not feasible, especially for complex functions and real-time applications. It simplifies the calculations by transforming the area under the curve into simple geometric shapes like rectangles and trapezoids.

The area under a curve is determined by integral calculus. You start by identifying the limits of integration, that is the "start" and "end" point on the x-axis. Then, you integrate the equation of the curve between these points. The resulting value is the area under the curve.

One can calculate the area under the curve without integration by using numerical methods such as the Trapezoidal Rule or Simpson's Rule. These methods involve dividing the area into many small sections, approximating each section as either a trapezoid or a parabola, then summing these areas.

The area under a curve can be estimated using numerical integration methods like the trapezoidal rule, Simpson's rule, or the rectangular method. It involves dividing the region under the curve into shapes (rectangles, trapezoids, or parabolas), computing their areas and then summing them up.

The area under a curve represents a quantity calculated by integrating the function that forms the curve. In engineering, it can depict various physical quantities such as displacement, total production, or energy, depending on the context of the graph.

Integration gives the area under a curve because it essentially adds up infinitesimally small slices of area under the curve. This is achieved by multiplying the height or value of the function at a particular point by an infinitesimally small width to compute the area of each slice.

What does the term 'Area under Curve' define in engineering mathematics?

'Area under Curve' defines the area bounded by the graph of a function and the x-axis. It helps quantify physical quantities like displacement, work, energy, and population predictions.

How do you calculate the area under a curve in engineering mathematics?

The area under a curve can typically be calculated using definite integrals. Given a function f(x) and an interval [a, b], the area A under the curve of f from a to b is given by A = ∫ from a to b f(x) dx.

What are some practical applications of the concept of 'Area Under Curve'?

Applications include calculating the distance travelled by an object in physics, determining the amount of work done or energy used in electrical engineering, and determining the total cost of production in economics.

What is the Riemann sum method used for in the calculation of the area under a curve?

The Riemann sum method is used to approximate the area under a curve by partitioning the interval into subintervals, which act as the base for rectangles. The heights of these rectangles are determined by the value of the function at a specific point within each subinterval.

What is the Trapezoidal Rule method used for in the calculation of the area under a curve?

The Trapezoidal Rule method is used to approximate the area under a curve by dividing the area under the curve into an even number of intervals of equal width, and fitting trapezoids in these intervals.

What is the Simpson's Rule method used for in the calculation of the area under a curve?

Simpson's Rule method is used to provide a higher-accuracy estimate for the area under a curve by fitting parabolic arcs to approximate each interval.

Already have an account? Log in

Open in App
More about Area under curve

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up to highlight and take notes. It’s 100% free.

Save explanations to your personalised space and access them anytime, anywhere!

Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

Already have an account? Log in