Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Fundamental Theorem of Calculus

Calculus

8. Definite Integrals

Fundamental Theorem of Calculus

Previous problemNext problem

2:12 minutes

Problem 5.3.43

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₋₂⁻¹ 𝓍⁻³ d𝓍

Verified step by step guidance

Step 1: Recall the Fundamental Theorem of Calculus, which states that if a function f(x) is continuous on [a, b] and F(x) is its antiderivative, then ∫ₐᵇ f(x) dx = F(b) - F(a).

Step 2: Identify the integrand, which is x⁻³ (or 1/x³). The goal is to find its antiderivative. Rewrite the integrand as x⁻³ for clarity.

Step 3: Compute the antiderivative of x⁻³. Using the power rule for integration, ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1. For x⁻³, n = -3, so the antiderivative becomes -1/(2x²).

Step 4: Apply the limits of integration, -2 and -1, to the antiderivative. Substitute x = -1 and x = -2 into the antiderivative expression, F(x) = -1/(2x²).

Step 5: Calculate the difference F(-1) - F(-2) to evaluate the definite integral. This step involves substituting the values and simplifying the result.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

Definite integrals represent the signed area under a curve between two specified limits on the x-axis. They are denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral provides a numerical result that quantifies the accumulation of quantities, such as area, over the interval [a, b].

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫[a, b] f(x) dx = F(b) - F(a). This theorem allows us to evaluate definite integrals by finding the antiderivative of the integrand and calculating its values at the limits of integration.

Antiderivatives

An antiderivative of a function f(x) is another function F(x) such that F'(x) = f(x). Finding an antiderivative is essential for evaluating definite integrals using the Fundamental Theorem of Calculus. For example, if f(x) = x^n, the antiderivative is F(x) = (x^(n+1))/(n+1) + C, where C is a constant. Understanding how to find antiderivatives is crucial for solving integral problems.

Watch next

Master Fundamental Theorem of Calculus Part 1 with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Multiple Choice

Suppose the graph of a continuous function $f$ is shown below, and the area between the graph of $f$ and the $x$ -axis from $x = 0$ to $x = 2$ is $3$ (above the $x$ -axis), and from $x = 2$ to $x = 4$ is $2$ (below the $x$ -axis). What is the value of the definite integral $\int_{0}^{4} f (x) d x$ ?

views

Textbook Question