Question

Consider the following integral

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(0) Select the graph of the region whose area is represented by the definite integral
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15
Y
05
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o
Maple Generated Plot
(b) Use a geometric formula to evaluate the integral

Answer 1

(a) Top-left graph

(b) 1/3

Explanation:

Think about what the graph of
`f(x)=(x)/(6)` looks like. It is a diagonal line with a positive slope of
`(1)/(6)`, so if you take the area under that line from 0 to 2, you get a triangle! Hence, the top-left graph is the correct graph.

Since the area is just a triangle, we know that our base is 2-0=2 and our height is 2/6 = 1/3, so, geometrically, the integral is:

`\displaystyle \int\limits^2_0 {(x)/(6)} \, dx=(2((2)/(6)))/(2)=(2)/(6)=(1)/(3)`

If you want to do it the standard way using antiderivatives:

`\displaystyle \int\limits^2_0 {(x)/(6)} \, dx=(1)/(12)x^2 \biggr|^2_0=(1)/(12)(2)^2-(1)/(12)(0)^2=(4)/(12)=(1)/(3)`

As you can see, we arrive at the same answer!