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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 = 10 x ( 2 + 5 x ) ' y = 0 , x = 1 , x = 2

To determine

To calculate: The exact area of region bounded by the graphs of equations y=10x(2+5x), y=0, x=1 and x=2.

Explanation

Given Information:

The provided equations are y=10x(2+5x), y=0, x=1 and x=2.

Formula used:

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

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

The General power differentiation Rule is;

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=10x(2+5x)

To find the exact area of region bounded by the graphs of equations y=10x(2+5x), y=0, x=1 and x=2, integrate equation y=10x(2+5x) with lower limit x=1 and higher limit x=2, which can be mathematically written as:

12ydx=1210x(2+5x)dx

Rewrite.

12ydx=10121x(2+5x)dx

Here, a=2, b=5, u=x and du=dx.

Substitute u for x, a for 2, b for 5, and du for dx.

12ydx=10121u(2+5u)dx

Use the above integration formula and solve the above integral as;

12ydx=10[1aln|ua+bu|]12

Substitute x for u, 2 for a and 5 for b.

12ydx=10[12ln|x2+5x|]12=5[ln|x2+5x|]12=5[ln|22+5(2)|ln|12+5(1)|]=5[ln|212|ln|17|]

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