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# Warmup: Definite integral properties (no graph)

Apply the properties of definite integrals to evaluate definite integrals.

## Problem 1

Given ${\int }_{-5}^{-1}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=4$, ${\int }_{3}^{5}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=9$, and ${\int }_{-5}^{5}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=10$, find the following:
a) ${\int }_{-1}^{3}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=$

b) ${\int }_{-1}^{5}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=$

c) ${\int }_{-5}^{-1}7f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=$

d) ${\int }_{5}^{-5}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=$

### Challenge

e*) ${\int }_{3}^{5}\left(f\left(x\right)+4\phantom{\rule{0.167em}{0ex}}\right)dx=$

## Problem 2

Given ${\int }_{-8}^{-2}g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=11$, ${\int }_{-8}^{0}g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=5$, and ${\int }_{-2}^{8}g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=-5$, find the following:
a) ${\int }_{-2}^{0}g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=$

b) ${\int }_{0}^{8}g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=$

c) ${\int }_{8}^{-8}10g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=$

### Challenge

d*) ${\int }_{0}^{8}\left(g\left(x\right)+x\right)dx=$

## Want to join the conversation?

• In the final challenge question, how do we know to use the graph of y = x?
(3 votes)
• The way I think about it is that a definite integral is asking for the area under the curve/graph of the function within the integral. For example, in most of the problems above, we're looking for the integral (area under the curve) of the function y=g(x). But when we need to split the integral into two in the last problem, we're left with the integral (area under the curve) of y=g(x) and the integral (area under the curve) of y=x, because x was on its own and can be considered the function by itself.
(3 votes)
• Is there any way you could do this one for me? I got stuck
Integral of e^2x * (e(x)+1)^1/2
Thank you
(0 votes)
• = 2/15 (e x + 1)^(3/2) (3 e x - 2) + constant
(3 votes)