Please help me out with my calculus problems. Thank you !

1. Fall season is upon us. and Megan couldn't be happier. She recently bought myoelium substrate to help cultivate some tasty mushrooms for coming weeks. A few days after she sets up the substrate, Megan notices one particular mushroom is towering over the rest. She decides to track the growth of this mushroom for a few days. Let mt) denote the rate at which mushroom grows. measured in centimeters per day, i days after she starts tracking the mushroom. The function mfi] is given by: 10063 mm = W where f 33 I]. When Megan began tracking the rate of growth, the mushroom was 12 centimeters tall. (a) (b) (c) (d) Write an expression involving a denite integral for the height of the mushroom 6 days after Megan started tracking the growth of the mushroom. You do not {yet} have to evaluate this integral. Estimate the height of the mushroom 6 days after Megan started tracking its height by using both TRAPM] and MIDH). Write out the terms of your sums, and include units. Find the exact height of the mushroom .3 days after Megan started tracking its growth. Show all of your work. Include units. Megan notices a beautiful little mushroom in her batch which seems to be growing in a very peculiar manner and decides to track its height as well. Its height in centimeters t days after it blooms is given by Mt) = HI i132: cost, where l] 5 t 5 4. 100 Find an integral which gives the average height of this mushroom over the rst 4 days after it blooms, and solve that integral. showing the steps in your process. Include units in your nal IlS'W'Cl'. 2. Consider the region in the try-plane shown below bounded by the eurves y = 1:3 3. y = 1. y = 5 and the right-half of the circle given by (a- 8:12 + (y 3)2 = Hi. {a} {b} {C} 3} | I l l p 2 4 6 8 ll] 'Write an expression involving one or more integrals for the area of the region. You may use the fact that the area of a circle is avg. where r is the radius of the circle. Do not evaluate any integrals in your expression. 'Write an expression involving one or more integrals for the volume of the solid generated by revolving the region about the :i:axis. Use slices parallel to the :i:axis to determine your expression. Do not evaluate any integrals in your expression. 'Write an expression involving one or more integrals for the volume of the solid generated by revolving the region about the :raxis. L'se slioes perpendicular to the 1'axis to determine your expression. Do not evaluate any integrals in your expression. Do you expect that the integrals you found in parts {ll} and (e) evaluate to give the same number? "Thy or why not? What were the advantages (if any) of taking the strategy in part (b)? What were the advantages [if any) of taking the strategy in part {e}? Which strategy did your team prefer? Write an expression involving one or more integrals for the volume of the solid whose base is the region ahme and whose cross-seetiOnal sliees perpendicular to the yaxis are squares. Do not evaluate any integrals in your expression