Choose the definite integral that represents the shaded area below. – In the realm of calculus, definite integrals hold a pivotal role in representing the area under a curve. As we delve into the concept of choosing the definite integral that encapsulates the shaded region below, we embark on a journey that unravels the intricate relationship between geometry and integration.
The essence of definite integrals lies in their ability to quantify the area bounded by a curve and the x-axis. By dissecting the region into infinitesimal rectangles and summing their areas, we arrive at the definite integral, which provides an exact numerical value for the shaded region.
Integral Representation of Shaded Area
The definite integral represents the limit of Riemann sums, which are approximations of the area under a curve. It provides a precise way to calculate the area of irregular shapes or regions with curved boundaries.
For a function f(x) defined on an interval [a, b], the definite integral of f(x) from a to b, denoted as ∫[a,b] f(x) dx, represents the net area between the graph of f(x) and the x-axis over the interval [a, b].
If the area lies above the x-axis, the integral is positive, indicating the net area. If the area lies below the x-axis, the integral is negative, indicating the net area below the x-axis.
For example, consider the shaded region below the curve y = x^2 from x = 0 to x = 2. The definite integral that represents the shaded area is:
∫[0,2] x^2 dx
Types of Definite Integrals
Proper Definite Integrals
A proper definite integral has both upper and lower limits that are finite numbers. It converges to a finite value or diverges to infinity.
Improper Definite Integrals
An improper definite integral has one or both limits as infinity or negative infinity. It may converge to a finite value, diverge to infinity, or oscillate without converging.
Convergence or divergence of an improper integral depends on the behavior of the integrand at the infinite endpoints.
Methods for Evaluating Definite Integrals: Choose The Definite Integral That Represents The Shaded Area Below.
Fundamental Theorem of Calculus
The fundamental theorem of calculus provides a powerful tool for evaluating definite integrals. It states that if f(x) is continuous on [a, b], then the definite integral of f(x) from a to b is equal to the net change in the antiderivative of f(x) over the interval [a, b].
Integration by Substitution, Choose the definite integral that represents the shaded area below.
Integration by substitution involves changing the variable of integration to simplify the integral. It is useful when the integrand contains a composite function.
Integration by Parts
Integration by parts is a technique used to integrate the product of two functions. It is particularly useful when one of the functions is the derivative of the other.
Trigonometric Substitution
Trigonometric substitution is a technique used to integrate trigonometric functions. It involves expressing the integrand in terms of trigonometric functions and then using trigonometric identities to simplify the integral.
Applications of Definite Integrals
Finding Volumes
Definite integrals are used to find the volume of solids of revolution. By integrating the cross-sectional area of the solid over the appropriate interval, the volume can be determined.
Finding Areas
Definite integrals are used to find the area of regions in the plane. By integrating the function that defines the region over the appropriate interval, the area can be determined.
Other Applications
Definite integrals have applications in various fields, including physics, engineering, and economics. They are used to find work, energy, moments of inertia, and probabilities, among other applications.
Clarifying Questions
What is the relationship between a definite integral and the area under a curve?
The definite integral of a function over an interval represents the exact area bounded by the curve of the function, the x-axis, and the vertical lines at the endpoints of the interval.
How do you choose the correct definite integral for a shaded region?
To choose the correct definite integral, identify the function that defines the upper boundary of the shaded region and the interval over which the function is integrated. The definite integral should have the form ∫[a,b] f(x) dx, where [a,b] is the interval and f(x) is the function.
What are some applications of definite integrals?
Definite integrals have numerous applications, including finding volumes of solids of revolution, calculating work done by a force over a distance, and determining the center of mass of a region.
