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

1. $\int _ { 4 } ^ { 2 } - f ( x ) d x$

   Hint (a):

   Notice that the bounds have flipped and the integrand is now negative.

2. $\int _ { 10 } ^ { 12 } ( f ( x - 8 ) + 4 ) d x$

   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: $f\left(x − 8\right)$.

   $\int_{10}^{12}f(x-8)dx=\int_{2}^{4}f(x)dx=10.$

   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.

3. $\int _ { 10 } ^ { 12 } f ( x + 8 ) d x$

   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.