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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Area of a Region In Exercises 45-50, use the integration table in Appendix C to find the exact area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results.

y = 1 9 x 2 ( 2 + 3 x ) ' y = 0 , x = 1 , x = 2

To determine

To calculate: The exact area of region bounded by the graphs of equations y=19x2(2+3x), y=0, x=1 and x=2.

Explanation

Given Information:

The provided equations are y=19x2(2+3x), y=0, x=1 and x=2.

Formula used:

The formula for the integral 1u2(a+bu)du is,

1u2(a+bu)du=1a(1u+baln|ua+bu|)+C

The general power differentiation Rule:

ddx[un]=nun1dudx

Calculation:

Consider u=x.

Differentiate the considered function with respect to x using power rule of differentiation.

du=dx

Consider the provided expression:

y=19x2(2+3x)

To find the exact area of region bounded by the graphs of equations y=19x2(2+3x), y=0, x=1 and x=2, integrate equation y=1(x2+0.25)3/2 with lower limit x=1 and higher limit x=2, which can be mathematically written as:

12ydx=1219x2(2+3x)dx

Rewrite.

12ydx=19121x2(2+3x)dx

Here, a=2, b=3, u=x and du=dx.

Substitute u for x, a for 2, b for 3, and du for dx.

12ydx=19121u2(a+bu)du

Use the formula 12 and solve the above integral as:

12ydx=19[1a(1u+baln|ua+bu|)]12

Substitute x for u, 2 for a and 3 for b as;

12ydx=19[12(1x+32ln|x2+3x|)<

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