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Chapter 5: Problem 91

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{8} \sqrt[3]{y} d y$$

Short Answer

Expert verified
Question: Evaluate the definite integral of the given function: \(\int_{1}^{8}\sqrt[3]{y} dy\). Answer: The value of the definite integral is \(9\).

Step by step solution

01

Rewrite the function

Rewrite the function to a power instead of a square root. The given function is \(\sqrt[3]{y}\). We can rewrite it as follows: $$y^{\frac{1}{3}}$$ Now we can proceed with finding the antiderivative.
02

Find the antiderivative

To find the antiderivative of the given function, apply the power rule. The power rule is: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ In this case, our function is \(y^{\frac{1}{3}}\), so \(n=\frac{1}{3}\). Apply the power rule to find the antiderivative: $$\int y^{\frac{1}{3}} dy = \frac{y^{\frac{1}{3}+1}}{\frac{1}{3}+1} + C = \frac{y^{\frac{4}{3}}}{\frac{4}{3}} + C = \frac{3}{4}y^{\frac{4}{3}}+C$$
03

Apply the Fundamental Theorem of Calculus

Now, apply the Fundamental Theorem of Calculus to evaluate the definite integral given in the exercise: $$\int_{1}^{8} \sqrt[3]{y} d y = \frac{3}{4}y^{\frac{4}{3}}\Big|_1^8$$
04

Substitute the limits of integration

Plug in the upper limit and lower limit into the antiderivative and find the difference: $$\frac{3}{4}(8^{\frac{4}{3}}) - \frac{3}{4}(1^{\frac{4}{3}})$$
05

Evaluate the integral

Evaluate the expression to get the final answer: $$\frac{3}{4}(8^{\frac{4}{3}}) - \frac{3}{4} = \frac{3}{4}(16) - \frac{3}{4} = \boxed{9}$$ The value of the definite integral is then: $$\int_{1}^{8} \sqrt[3]{y} d y = 9$$

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Most popular questions from this chapter

Use the Fundamental Theorem of Calculus, Part \(1,\) to find the function \(f\) that satisfies the equation $$\int_{0}^{x} f(t) d t=2 \cos x+3 x-2$$ Verify the result by substitution into the equation.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that \(f, f^{\prime},\) and \(f^{\prime \prime}\) are continuous functions for all real numbers. a. \(\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))^{2}+C\) b. \(\int(f(x))^{n} f^{\prime}(x) d x=\frac{1}{n+1}(f(x))^{n+1}+C, n \neq-1\) c. \(\int \sin 2 x d x=2 \int \sin x d x\) d. \(\int\left(x^{2}+1\right)^{9} d x=\frac{\left(x^{2}+1\right)^{10}}{10}+C\) e. \(\int_{a}^{b} f^{\prime}(x) f^{\prime \prime}(x) d x=f^{\prime}(b)-f^{\prime}(a)\)

Consider the function \(f(x)=x^{2}-4 x\) a. Graph \(f\) on the interval \(x \geq 0\) b. For what value of \(b>0\) is \(\int_{0}^{b} f(x) d x=0 ?\) c. In general, for the function \(f(x)=x^{2}-a x,\) where \(a>0,\) for what value of \(b>0\) (as a function of \(a\) ) is \(\int_{0}^{b} f(x) d x=0 ?\)

General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f\). Assume \(f\) and its derivatives are continuous for all real numbers. $$\int\left(5 f^{3}(x)+7 f^{2}(x)+f(x)\right) f^{\prime}(x) d x$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\frac{1}{2} \int_{0}^{\ln 2} e^{x} d x$$
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