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Chapter 6: Q. 24 (page 510)

Each of the definite integrals in Exercises 19鈥�24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.

蟺鈭�04(22-y2)dy

Short Answer

Expert verified

The required region is shown as,

Step by step solution

01

Step 1. Given Information     

The given integral is蟺鈭�04(22-y2)dy

02

Step 2. Explanation  

The given integral is of the form 蟺鈭�ab(f(y)2)dy which represents the volume of a solid formed by rotating the region bound by f(y) and y-axis in the interval [a,b] around y-axis.

On comparing the given region with the above region, we have,

f(y)=22-(y)2=4-yintheinterval[0,4]

Express f(y) as x and rewrite the function,

x=4-y4-y=x2y=4-x2

The simplified function represents a parabola.

Plot the graph of the function and identify the region between the intervals.

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