Chapter 6: Q. 18 (page 510)

For each pair of definite integrals in Exercises 13–18, decide which, if either, is larger, without computing any integrals.
$π \int_{0}^{4} ydy and π \int_{4}^{8} (8 - y) dy$

Short Answer

Expert verified

The value of both the integrals are equal.

Step by step solution

Step 1. Given Information

The given integrals are $π \int_{0}^{4} ydy and π \int_{4}^{8} (8 - y) dy$

Step 2. Explanation

The integrals are of the form $π \int_{a}^{b} (f {(y))}^{2} dy$ which represents the volume of the solid formed by rotating the region bound by f(y)and y-axis in the interval $[a, b]$ around x-axis.

On comparing the given integrals with the above given relation and observed that the function involved are $f (y) = \sqrt{y} in the interval [0, 4] and f (y) = \sqrt{8 - y} in the interval [4, 8]$

Now, we will plot the graph of both the functions and identify the regions between the interval.

From the graph, we can say that both the regions are identical and symmetrical. This means the volume of the solids formed by rotating these regions will be equal. Thus, the value of both the integrals is also equal.

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For each pair of definite integrals in Exercises 13–18, decide which, if either, is larger, without computing any integrals.
$π \int_{0}^{4} ydy and π \int_{4}^{8} (8 - y) dy$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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For each pair of definite integrals in Exercises 13–18, decide which, if either, is larger, without computing any integrals.π∫04ydyandπ∫48(8-y)dy

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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Most popular questions from this chapter

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For each pair of definite integrals in Exercises 13–18, decide which, if either, is larger, without computing any integrals.
$π \int_{0}^{4} ydy and π \int_{4}^{8} (8 - y) dy$