   Chapter 14, Problem 10RE

Chapter
Section
Textbook Problem

Finding the Area of a RegionIn Exercises 7–10, use an iterated integral to find the area of the region bounded by the graphs of the equations. x = y 2 + 1 ,     x = 0 ,     y = 0 ,     y = 2

To determine

To calculate: The area of the region bounded by the graphs of given equation x=y2+1, x=0, y=0 and y=2.

Explanation

Given: The provided equations are:

x=y2+1, x=0, y=0 and y=2

Formula used: The area of a region is given by

A=xlxhylyhdydx

Vertex form of parabola’s equation is given by

(xh)=a(yk)2

Here (h,k) is the vertex of the parabola

Calculation: The equation x=y2+1 is the equation of a parabola.

Write it in its vertex form

x=y2+1x1=y2(x1)=(y0)2

Vertex of this parabola is given by

V=(1,0)

The equation x=0 represents y-axis.

The equation y=0 represents x-axis.

The equation y=2 is a straight line parallel to x-axis passing through y=2.

Intersection of parabola and line y=2 is:

x=y2+1=22+1=5

Therefore, the intersection point is (5,2)

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