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

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

Integrating with Respect to y
Section 7.2b

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

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

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

Quality Practice Problems
Try #4 from p.380:

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)

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)

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:

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

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)

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)

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

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