   Chapter 13.2, Problem 49E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 49-54, use properties of definite integrals. 52.  If ∫ 1 2 ( 2 x − x 2 )  d x = 2 3 ,  and  ∫ 2 4 ( 2 x − x 2 ) d x = − 20 3 , what does  ∫ 4 1 ( x 2 − 2 x )   d x  equal ?

To determine

To calculate: The value of the integral 14(x22x)dx if 12(2xx2)dx=23 and 24(2xx2)dx=203.

Explanation

Given Information:

The value of integral 12(2xx2)dx=23 and 24(2xx2)dx=203.

Formula used:

The integral abf(x)dx can be written as acf(x)dx+cbf(x)dx where acb.

The integral abcf(x)dx can be written as cabf(x)dx where c is any real number.

Calculation:

Consider the value of the integrals 12(2xx2)dx=23 and 24(2xx2)dx=203.

Recall that the integral abf(x)dx can be written as acf(x)dx+cbf(x)dx where acb.

Since, 124.

Thus, 14(2xx2)dx=12(2xx2)dx+24(2xx2)dx.

Substitute 23 for 12(2xx2)dx and 203 for 24(2xx2)dx in 14(2xx2)dx=12(2xx2)dx+24(2xx2)dx

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