1. Oregon Department of Fish and Wildlife asks you to estimate the surface area and volume...

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1. Oregon Department of Fish and Wildlife asks you to estimate the surface area and volume of Paulina Lake. They give you a map of Paulina Lake with length and width dimensions as well as various depth soundings across the lake (in purple). (a) Estimate the area of the surface of the lake. (Note this is a real problem, so there are multiple approaches. Clearly indicate any assumptions you make in your approach.) (b) Estimate the volume of the lake using trapezoid shaped cross-sections where you assume the bottom of the trapezoid is half the length of the top. (c) What other shape of cross-sections might you use to estimate the volume of the lake? How would that change your calculation? (120 H) ft 9000 5000 ft (240 ft) Good fr (ઞs ++) 91004+ (vs+) 9400 ft 8200 ft (210 ft) 1800 ft (100) (depths) 2. Consider the region between y = ln x, y = 0 and x = 2 and the solid found by rotating this region around the line x = 3. (a) Set up an integral to calculate the volume of this solid using washers. (Which way do you slice the region? Draw a picture and explicitly indicate on the graph the important radii.) (b) Set up an integral to calculate the volume of this solid using shells. (Which way do you slice the region? Draw a picture and explicitly indicate on the graph the important radius and height.) (c) Determine which of the above two integrals you can evaluate by hand and evaluate that integral by hand to determine the volume.¹ 1. Oregon Department of Fish and Wildlife asks you to estimate the surface area and volume of Paulina Lake. They give you a map of Paulina Lake with length and width dimensions as well as various depth soundings across the lake (in purple). (a) Estimate the area of the surface of the lake. (Note this is a real problem, so there are multiple approaches. Clearly indicate any assumptions you make in your approach.) (b) Estimate the volume of the lake using trapezoid shaped cross-sections where you assume the bottom of the trapezoid is half the length of the top. (c) What other shape of cross-sections might you use to estimate the volume of the lake? How would that change your calculation? (120 H) ft 9000 5000 ft (240 ft) Good fr (ઞs ++) 91004+ (vs+) 9400 ft 8200 ft (210 ft) 1800 ft (100) (depths) 2. Consider the region between y = ln x, y = 0 and x = 2 and the solid found by rotating this region around the line x = 3. (a) Set up an integral to calculate the volume of this solid using washers. (Which way do you slice the region? Draw a picture and explicitly indicate on the graph the important radii.) (b) Set up an integral to calculate the volume of this solid using shells. (Which way do you slice the region? Draw a picture and explicitly indicate on the graph the important radius and height.) (c) Determine which of the above two integrals you can evaluate by hand and evaluate that integral by hand to determine the volume.¹