Math191_Week3_section - 5 m at the rate of . 2 m 3 / sec....

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Math 191 Problem Set for Week 3 Section 1. Evaluate the integrals in (a) and (b). a. R (2 r - 1) cos 3(2 r - 1) 2 +6 3(2 r - 1) 2 +6 dr b. R sin θ θ cos 3 θ 2. R 4 1 dy 2 y (1+ y ) 2 3. Find the area of the region enclosed by y = x 4 - 4 x 2 + 4 and y = x 2 . 4. Find the volume of the solid generated by revolving the region bounded by the parabola y = x 2 and the line y = 1 about the line y = - 1 . 5. Volume of a bowl a. A hemispherical bowl of radius a contains water to a depth h . Find the volume of water in bowl. R ( y ) = p a 2 - y 2 V = π R - a + h - a ( a 2 - y 2 ) dy = π ± a 2 y - y 3 / 3 ² - a + h - a = π [ a 2 h - a 3 - ( h - a ) 3 / 3 - ( - a 3 + a 3 / 3)] = π [ a 2 h - 1 / 3( h 3 - 3 h 2 a + 3 ha 2 - a 3 ) - a 3 / 3] = π ( a 2 h - h 3 / 3 + h 2 a - ha 2 ) = πh 2 (3 a - h ) / 3 b. Water runs into a sunken concrete hemispherical bowl of radius
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Unformatted text preview: 5 m at the rate of . 2 m 3 / sec. How fast is the water level in the bowl rising when the water is 4 m deep? Given dV dh = 0 . 2m 3 / sec and a = 5m, nd dh dt | h =4 . From part (a), V ( h ) = h 2 (15-h ) / 3 = 5 h 2-h 3 / 3 dV dh = 10 h-h 2 dV dt = dV dh dh dt = h (10-h ) dh dt dh dt | h =4 = . 2 4 (10-4) = 1 (20 )(6) = 1 120 m/sec. 1...
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This note was uploaded on 06/20/2008 for the course MATH 1910 taught by Professor Berman during the Spring '07 term at Cornell University (Engineering School).

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