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6.1 - Math 182 Homework Section 6.1 Problem 10 Sketch the...

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Unformatted text preview: Math 182 Homework Section 6.1 Problem 10 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y . Draw a typical rectangle and then find the total area of the region. Region bounded by y = x 2 and y = 4 x- x 2 The curves intersect when x 2 = 4 x- x 2 2 x 2- 4 x = 0 2 x ( x- 2) = 0 x = 0 or x = 2 When x = 0 we have that y = 0 2 = 4 - 2 = 0 and when x = 2 it follows that y = 2 2 = 4 2- 2 2 = 4 so the points of intersection are (0 , 0) and (2 , 4) as shown below: The width of the infinitely thin rectangles is dx and their height is the difference in y- coordinates given by y down parabola- y up parabola = 4 x- x 2- x 2 = 4 x- 2 x 2 . Therefore, the total area is integraldisplay 2 (4 x- 2 x 2 ) dx = bracketleftbigg 2 x 2- 2 3 x 3 bracketrightbigg | x =2 x =0 = 2 2 2- 2 3 2 3- (0 2- 2 3 3 ) = 8- 16 3 = 24- 16 3 = 8 3 Problem 16 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y .....
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6.1 - Math 182 Homework Section 6.1 Problem 10 Sketch the...

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