Question

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

Answer 1

Answer:

(a) Top-left graph

(b) 1/3

Step-by-step explanation:

Think about what the graph of $f(x)=\frac{x}{6}$ looks like. It is a diagonal line with a positive slope of $\frac{1}{6}$, 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:

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

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

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

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