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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.4.78a
Chapter 8, Problem 8.4.78a

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
a. Find the area of the region bounded by the graph of f and the x-axis on the interval [0,2].

Verified step by step guidance
1
Identify the function and the interval for which you need to find the area. Here, the function is \(f(x) = \frac{x^2}{3}\) and the interval is \([0, 2]\).
Recall that the area under the curve of a function \(f(x)\) from \(a\) to \(b\) is given by the definite integral \(\int_a^b f(x) \, dx\). In this case, you want to compute \(\int_0^2 \frac{x^2}{3} \, dx\).
Set up the integral explicitly: \(\int_0^2 \frac{x^2}{3} \, dx = \frac{1}{3} \int_0^2 x^2 \, dx\). You can factor out the constant \(\frac{1}{3}\) from the integral.
Find the antiderivative of \(x^2\), which is \(\frac{x^3}{3}\). So, the integral becomes \(\frac{1}{3} \left[ \frac{x^3}{3} \right]_0^2\).
Evaluate the definite integral by substituting the upper and lower limits: calculate \(\frac{1}{3} \left( \frac{2^3}{3} - \frac{0^3}{3} \right)\) to express the area bounded by the graph of \(f\) and the x-axis on \([0,2]\).

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Key Concepts

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

Definite Integral as Area Under a Curve

The definite integral of a function over an interval represents the net area between the graph of the function and the x-axis. For a non-negative function, this integral gives the exact area bounded by the curve and the x-axis within the specified limits.
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Definition of the Definite Integral

Integration of Polynomial Functions

Integrating polynomial functions involves applying the power rule, which states that the integral of x^n is (x^(n+1))/(n+1) plus a constant. This rule simplifies finding antiderivatives for functions like f(x) = x²/3.
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Taylor Polynomials

Evaluating Definite Integrals Using Limits

To compute the definite integral, first find the antiderivative, then evaluate it at the upper and lower bounds of the interval. Subtracting these values yields the exact area under the curve between those points.
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Definition of the Definite Integral
Related Practice
Textbook Question

Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.

a. Use the reduction formula ∫ from 0 to ∞ of x^p e^(-x) dx = p ∫ from 0 to ∞ of x^(p-1) e^(-x) dx for p = 1, 2, 3, ...

to show that Γ(p + 1) = p! (p factorial).

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Textbook Question

63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.

Textbook Question

65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

70. Let f(x) = e^(-x²).

a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.

Textbook Question

41-44. {Use of Tech} Nonuniform grids

Use the indicated methods to solve the following problems with nonuniform grids.

41. A curling iron is plugged into an outlet at time t = 0. Its temperature T in degrees Fahrenheit, assumed to be a continuous function that is strictly increasing and concave down on 0 ≤ t ≤ 120, is given at various times (in seconds) in the table.

a. Approximate (1/120)∫(0 to 120)T(t)dt in three ways using a left Riemann sum, using a right Riemann sum and using the Trapezoid Rule

Interpret the value of (1/120)∫(0 to 120)T(t)dt in the context of this problem.

Textbook Question

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.