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Chapter 11: Problem 19

Find the indefinite integral and check your result by differentiation. $$ \int y^{3 / 2} d y $$

Short Answer

Expert verified
The indefinite integral \(\int y^{3 / 2} d y\) is \(2/5 y^{5/2} + C\). After differentiating this result, we indeed get back the original function \(y^{3 / 2}\).

Step by step solution

01

Compute the Integral

Applying the power rule for integration to \(\int y^{3/2}dy\), we have the integral as \(1/(3/2+1) y^{3/2+1} + C = 2/5 y^{5/2} + C \)
02

Check the Result Using Differentiation

Now differentiate the result to check if the differentiated function matches the integrand. Differentiating \(2/5 y^{5/2} + C\) using the power rule yields \(1/2 * 5/2 y^{5/2-1} = y^{3/2}\). This matches the original integrand, so the integral is correct
03

Confirm the Answer

The indefinite integral of \(y^{3/2}\) is \(2/5 y^{5/2} + C\) and the differentiation of the integral gives back the original function \(y^{3/2}\).

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Most popular questions from this chapter

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{1}{x}, y=x^{3}, x=\frac{1}{2}, x=1 $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\sqrt{x}, \quad[0,1] $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d C}{d x}=2.25 \quad x=100 $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=\frac{1}{(x-3)^{2}} \quad[0,2] $$
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