(1 Convert the integral below to polar coordinates and evaluate the integral. ∫4/2√0∫16−y2√yxydxdy∫04/2∫y16−y2xydxdy Instruction
s: Please enter the integrand in the first answer box, typing theta for θθ. Depending on the order of integration you choose, enter dr and dtheta in either order into the second and third answer boxes with only one dr or dtheta in each box. Then, enter the limits of integration and evaluate the integral to find the volume. ∫BA∫DC∫AB∫CD =_____A = _______B = _______C = _______D =_______Volume =_______
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So.. if you take a peek at the picture below
the trunk is really just a half-cylinder on top of a square, with a depth of 2 meters
what's the volume? well, easy enough, take the volume of the cylinder, then half it
take the volume of the rectangular prism, and then add them both
now.. for the surface area...
now.. for the surface area of the prism... well
is really just 6 rectangles stacked up to each other at the edges
so... get the area of the lateral rectangles, and the one at the bottom, skip the rectangle atop, because is the one overlapping the cylinder, and is not outside, and thus is not surface area then
for the lateral ones, you have a front of 1x1, a back of 1x1 and a left of 1x2 and a right of 1x2, and then the one at the bottom, which is a 1x2
then add both surface areas, and that's the surface area of the trunk
the trunk is really just a half-cylinder on top of a square, with a depth of 2 meters
what's the volume? well, easy enough, take the volume of the cylinder, then half it
take the volume of the rectangular prism, and then add them both
now.. for the surface area...
now.. for the surface area of the prism... well
is really just 6 rectangles stacked up to each other at the edges
so... get the area of the lateral rectangles, and the one at the bottom, skip the rectangle atop, because is the one overlapping the cylinder, and is not outside, and thus is not surface area then
for the lateral ones, you have a front of 1x1, a back of 1x1 and a left of 1x2 and a right of 1x2, and then the one at the bottom, which is a 1x2
then add both surface areas, and that's the surface area of the trunk
