CPM Homework Help : CALC Problem 4-88

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4-88.

Rewrite the following integral expressions as single integrals.

$\int _ { - 3 } ^ { - 5 } f ( x ) d x + \int _ { - 5 } ^ { - 3 } g ( x ) d x$
Hint (a):
$\int_{a}^{b}f(x)dx=\int_{b}^{a}-f(x)dx$
Answer (a):
$\int_{-5}^{-3}\left (g(x)-f(x) \right )dx$

$3 \int _ { 1 } ^ { 6 } f ( x ) d x + 5 \int _ { 1 } ^ { 6 } g ( x ) d x$
Hint (b):
Translation: three times the area under $f(x)$ between $x = 1$ and $x=6+$ five times the same area. Write this as a single integral.

$\int _ { 6 } ^ { 11 } f ( x ) d x + \int _ { 11 } ^ { 6 } f ( x ) d x$
Hint (c):
See hint (a) and then see hint (c).

$\int _ { 7 } ^ { 10 } f ( t ) d t - \int _ { 7 } ^ { 9 } f ( t ) d t$
Hint (d):
Since an integral is the area underneath or above a curve, if you take a smaller portion of that area and subtract it from a larger portion, both starting at the same point, what are you left with?

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