1. Do Now: p.381: #8
Integrate the two parts separately:
Shaded Area =

2. Integrating with Respect to y
Section 7.2b

3. Integrating with Respect to y
Sometimes the boundaries of a region are more easily
described by functions of y than by functions of x. In such
cases, we can use horizontal rectangles: d d c c

4. Returning to an example from last class…
Find the area of the region R in the first quadrant that is bounded
above by and below by the x-axis and the line
Solve our previous
equations for x: 2
(4,2) 1 2 4

5. Returning to an example from last class…
And a third way to handle this example: Using geometry formulas.
(4,2) 2
Integrate the square root
function from 0 to 4, then
subtract the area of the
triangle on the right: 1 2 2 2 4

6. Quality Practice Problems
Try #4 from p.380:

7. Quality Practice Problems
Find the area of the region enclosed by the given functions.
How about the graph?
To find the y-coordinates of the
Intersection points, solve:
(4,2)
(1,–1)

8. Quality Practice Problems
Find the area of the region enclosed by the given functions.
Use your calculator to graph in [–3, 3] by [–3, 3].
We could integrate with respect to x,
but that would require splitting the
region at x = a.
(c,d)
Instead, let’s integrate from y = b to
y = d, handling the entire region at
once…
(a,b)

9. Quality Practice Problems
Find the area of the region enclosed by the given functions.
Use your calculator to graph in [–3, 3] by [–3, 3].
Calculate b and d, storing the values
in your calculator:
(c,d)
(a,b)
Evaluate the area numerically:

10. Do Now: p.381: #8
Returning to the “Do Now,” let’s use our new technique!!!
Shaded Area =

11. Quality Practice Problems
Find the area of the region between the curve and
the line by integrating with respect to (a) x, (b) y.
Integrate with respect to x:
Graph the region:
(–2,–1)
(2,–1)

12. Quality Practice Problems
Find the area of the region between the curve and
the line by integrating with respect to (a) x, (b) y.
Integrate with respect to y:
Graph the region:
(–2,–1)
(2,–1)

13. Find the area of the region in the first quadrant bounded on the
left by the y-axis, below by the line , above left by the
curve , and above right by the curve
Graph the region:
Integrate in two parts:
x = 1
x = 4

14. Find the area of the region in the first quadrant bounded on the
left by the y-axis, below by the line , above left by the
curve , and above right by the curve
Graph the region:
x = 1
x = 4