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Asked by Alex

Sketch the functions y1 = - x 2 and y2 = x2 - 2 on an x-y system of axes and using integral calculus determine the area enclosed by them. Then: The area is now rotated 360degrees about the x-axis. Using integral calculus determine the volume?

Hello, this app does not allow me to take photo or attach video. However, I am willing to help. To calculate volume, we need to identify the intersection points. y1 = y2 -x^2 = x^2 - 2 2x^2 = 2 x = +\- 1 This yields y = -1. When you sketch the graph, you will have 2 quadratic curves with upwards and downwards. The calculated one is the intersection between the two curves We then have a define formula of the integral at the y-axis V = \pi \int x^2 dy Hence, we can calculate the volume as V = \pi \int_{-2}^{0} x_{1}^{2} - x_{2}^{2} dy = \pi \int_{-2}^{0} (-y^2)^{2} - (y+ 2)^{2} dy = \pi \int_{-2}^{0} y^{4}- (y+ 2)^{2} dy = \pi [y^5/5 - (y+2)^{3}/3]_-2 ^ 0 = pi 24/3 This is your answer. Once again, I wish this app allowed the use of photo and videos for better explanation. Thank you

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Keng Han Lee
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