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4-77.

If $\int _ { 2 } ^ { 4 } f ( x ) d x=10$ $\int _ { 2 } ^ { 4 } f ( x ) d x = 10$ find:

$\int _ { 4 } ^ { 2 } - f ( x ) d x$
Hint (a):
Notice that the bounds have flipped and the integrand is now negative.

$\int _ { 10 } ^ { 12 } ( f ( x - 8 ) + 4 ) d x$
Step 1(b):
Split this problem into a sum of two integrals:
$\int_{10}^{12}f(x-8)dx+\int_{10}^{12}4dx$
Step 2(b):
Integral 1: Notice that the bounds shifted 8 units to the right... but so did the integrand: $f(x − 8)$ .
$\int_{10}^{12}f(x-8)dx=\int_{2}^{4}f(x)dx=10.$
Step 3(b):
Integral 2: This integral represents a rectangular area with base: $12-10 = 2$ and height: $4$ .
$\int_{10}^{12}4dx=(2)(4)=8$
Step 4(b):
Add.

$\int _ { 10 } ^ { 12 } f ( x + 8 ) d x$
Hint (c):
The bounds shifted $8$ units to the right, but the integrand shifted $8$ units to the left.
Answer (c):
We do not have enough information to evaluate this integral.

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