Please answer the following :( !!! I will rate you 5 starssss!!! Consider the ar

Question

Please answer the following :( !!! I will rate you 5 starssss!!!

Consider the area shown below. The curve drawn is x2 + y2 = 2, and we have used the notation Dy for Deltay. (Click on the figure for a larger version.) Write a Riemann sum for the area, using the strip shown: Riemann sum = (sqrt(2-y^2))Dy Now write an integral that gives this area (sqrt(2-y^2))dy where a = 0 and b = Finally, calculate the exact area of the region, using your integral area = Consider the area shown below. The top curve (in blue) is y = , and the bottom curve (in red) is y = x3, and we have used the notation Dy for Deltay. (Click on the figure for a larger version.) Write a Riemann sum for the area, using the strip shown: Riemann sum = Now write an integral that gives this area where a = 0 and b = 1 Finally, calculate the exact area of the region, using your integral area = Consider the volume of the region shown below, which shows a hemisphere of radius 7 mm and a slice of the hemisphere with width Dy = Deltay. Write a Riemann sum for the volume, using the slice shown: Riemann sum = Now write an integral that gives this volume where a = and b = Finally, calculate the exact volume of the region, using your integral volume = (include units)

Explanation / Answer

first box:

b = sqrt(2)

area: the definite integral you wrote out, i'll let you do that =)

second box:

i don't know if you NEED to use Dy, but if not its sqrtx - x^3 dx

if so, x = y^2 and x = cube root y, so riemann sum is: y^2 - cube root y dy

(i don't know how you would type in cube root so i think you need to use dx)

use that again for integral

area = plug 1 in for x, so 2/3(1)^3/2 - (1)^4/4 = 2/3 - 1/4 = .4167

third box:

area of hemisphere = 2pi*r^2, and y is r in this case

2pi*y^2 dy is your riemann sum and integral

it is from 0 to your radius, which is 7

integrated, becomes 2/3*pi*y^3 over 0 to 7

comes out to 718.38

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