# Solutions to Problem Set 11

Math 211-03

9-26-2017

[Area]

1. Find the area of the region bounded by and

Find the intersections:

The curves intersect at and . Between 1 and 5, the top curve is and the bottom curve is . The area is

2. Find the area of the region between and from to .

The lines cross in the middle of the interval. To find the intersection point, solve simultaneously:

The lines cross at .

From to , the line is the top curve and the line is the bottom curve. From to , the line is the top curve and the line is the bottom curve. The area is

3. Find the area of the region between the curves and from to .

The curves cross in the middle of the interval. To find the intersection point, solve simultaneously:

This gives and . The intersection point between 0 and 3 is at .

I'll use vertical rectangles to compute the area.

For the left-hand piece, the top curve is and the bottom curve is .

For the right-hand piece, the top curve is and the bottom curve is .

The total area is

4. Find the area of the shaded region in the picture below.

It's possible to compute the area using vertical rectangles, but you'll need two integrals to do it. I'll use horizontal rectangles instead.

The thickness of a horizontal rectangle is , so I want to express my integrand in terms of y.

The right-hand end of a horizontal rectangle lies on , which is . The left-hand end lies on , which is .

Set the curves equal to find the intersection point:

When , . So the limits on y are (the x-axis) and (the intersection point). The area is

5. Find the area of the region bounded by and .

Use horizontal rectangles. The right-hand end of a horizontal rectangle lies on and the left-hand end lies on .

Find the intersection points:

The area is

Our experience is composed rather of illusions lost than of wisdom acquired. - Joseph Roux

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