Evaluate each integral

Question

The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral.

(a)    6	g(x) dx
0

 

(b)    18	g(x) dx
6

 

(c)    21	g(x) dx
0

 

Answer 1

So, we COULD figure out the equations for these pieces of the graph and evaluate the integrals. But remember, the definite integral of a function/graph tells you the area of the curve (i.e. the area between g and the x-axis). So we also could just use geometry to get these areas, and that will give the same answer as doing the integrals. This method will be much less annoying.

For example, the first integral is just the integral of the first linear bit. But notice, that linear bit creates a triangle, with the x and y axes forming the two legs of the right triangle. The base of the triangle is 6, and the height is 12, and so using the faction that the area of a triangle is 1/2 base*height, the first integral is just

0.5*6*12=36.

The second integral is the circle bit. Specifically, this is a semi-circle with radius 6 BELOW the x-axis, meaning the area will be negative and 1/2 that of a full circle. That being said, the evaluation of the integral is

-(1/2)pi*6²= - 18pi

We're not asked it directly, but we'll need it for the last part, so let's do the integral of g from 18 to 21. Thisis the last linear bit, another triangle. This one has base 3 and height 3, so the integral is

1/2 3*3=9/2

The last integral you're being asked is the integral of the function g over the entire domain we've addressed. This is just the sum of the areas we've calculated.

36+-18pi+9/2=81/2-18pi