This function, graphed on [-5,2], produces two rectangles:
We count the little rectangle (square) above the x-axis positively, and the big one below the x-axis negatively:
Find the integral.
Now we have two triangles:
First we need to figure out where the line hits the x-axis, since that will tell us what the bases of the triangles are.Since
when x = 4, we know x = 4 is the point in between the two triangles.
Now we can figure out the necessary dimensions:
And find the integral:
Find the integral. (hint: draw pictures)
don't try to calculate any areas
The hint says to not try to calculate any areas. Since the sin function is odd, the area below the x-axis and the area above the x-axis are equal:
When we count one area positively and one area negatively, the weighted areas cancel out. So
Find the integral.(hint: draw pictures)
calculate the missing area
describes half of a circle sitting on top of the x-axis. The area we want is the area inside the purple 2-by-1 rectangle, but not inside the half-circle:
The rectangle has area 2 and the half-circle has area , so
For a sanity check, this is approximately 0.429, which is positive, as would be expected when finding the integral of a non-negative function.
Find the integral.
where g(t) = 3t + 4.
Here we have another triangle:
If and a is positive, what is a?
This graph consists of two equally sized triangles:
The triangle on the right has area . Since the triangles have the same size, together they have area
So we know . We're given , so 49 and a2 must be the same. That means a = 7.
If and a is positive, find a.
This is like the previous problem, only easier! From a graph we can see that the region between the graph of x and the x-axis is a triangle with area .
9 = a2
a = 3.
We only keep the answer a = 3 since we were told a was positive.
Find c given that .
Since the integral is positive, we can assume that c is positive. If we graph the constant c on [-10,10] we get a rectangle of height c and length 20. This means the area of the rectangle is 20c. This must equal 40, which we were told was the value of the integral, so c = 2.