I need help with this homework. I know it is a lot of questions but I would appreciate it if you can help me with as many as possible you can. Thanks. MTH 264, Homework #1, Summer 2022 NameInstructions: Write your work up neatly and attach to this page. Record your final answers (only)directly on this page if they are short; if too long indicate which page of the work the answer is on andmark it clearly. Use exact values unless specifically asked to round.1.Find the area bounded by the curves and sketch the region.a. f(x)=x² - 4x² and g(x)=x² - 4xb.y = x² - 2x, y = x +4c. y = cos x, y = 2d. y = x³, y = xe. y = |x|, y = x² - 2cos x, [0, 2π]2. Set up the integral to find the value of the area bounded by the graphs. You do not need toevaluate it, but do sketch the region.a.y =21+x4, y = x²b. y = cos x, y = x + 2 sin4 xi.ii.y-axisthe line y = 83. Calculate the volumes of the solids of revolution as indicated below. Calculate the volume bothwith the shell method and with the disk or washer method and verify that the volume is thesame in each case.a. y=2x², y=0, x=2 Revolved around the:10b. y=₁y=0,₂x = 1, x = 5 Revolved around:i.y-axisy = 10ii.c. y = ex, y = 1, x = 2 Revolved around:i.y-axisx-axisii.d. x = y², x = 1 - y² Revolved around:i.y = 1e. y = x³, y = 0, x = 1 Revolved around:i.ii.y-axisx-axis4. Find the surface area when the function y=x³1.3f. y = ex, y = x² − 1, x = −1, x = 1g. x = 1 - y², x = y² - 1h. x = y¹, y = √2x, y = 01Xi.y ==, y = x, y = x, x 0j. y = x²e-x, y = xe¯xc. y = e¹-x²1, y=x4+6 2xiii. x-axisiv. the line x = 2iii. x-axisiv. x = 5iii. y = 0iv. x = 3ii. x = 3iii. x = 1iv. y = 1on the interval 1≤x≤2.5. Find the area of the surface generated by revolving the curve about i) the x-axis, ii) the y-axis.Sketch the graph. You may need to integrate numerically.a. x=t, y=4-2t [0,4]d. x = a cos 0, y = asin0 [0,2π]b. x = t³, y = t², [0,1]e. x = t cost, y = t sint, [0,1] C. x = sint, y = sin 2t,f. x = et ―t, y = 4et/2, [0,1]6. A torus is formed by revolving the region bounded by the circle x² + y² = 4 around the linex=3. Find the volume of this doughnut-shaped solid. [Hint: find the function that represents thesemicircle.]7. A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of12-x and the x-axis, around the x-axis, where x and y are measured in meters. Findthe tank's volume.x²818. An ornamental light bulb is designed by resolving the graph of y=x2-x/² on the interval3[0,1/3] around the x-axis, where x and y are measured in feet. Find the surface area of the bulband use the result to approximate the amount of glass needed to make the bulb (assume thatthe glass is 0.015 inches thick).9. Find the volume of the region bounded by y = ex, y = ex, x = 1 around the y-axis. (Useshells.)10. Differentiate.a.[0,1]F(x) = [ - dttb.h(x) = f*t cos(t²) dtSoc. y(x) = ex² e-t² dtd. u(t)= ecos(sin² 3t)e. q(t)cos(Int)tf. v(t) = tan=3a. y = 1 + 6x², [0,1]b. y = (1-e-*), [0,2]x3c. y = ¹+1,[1,2]4xd.3y = x sinx, [0,2π]g. g(x)=√t + sint dt-x eth. r(x) = f == dt√√xti.a(t) = 2t-1j. p(x)k.s(t)11. Find the length of arc in rectangular coordinates. You may need to calculate the valuenumerically.√x² + 3arcsin(et + t)e. y = In(secx), [0,7]3f. y = √√x, [0,1]g. y = √x - x² + arcsin x, [0,1]h. y = e-x²,[1,1]

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I need help with this homework. I know it is a lot of questions but I would appreciate it if you can help me with as many as possible you can. Thanks. MTH 264, Homework #1, Summer 2022 NameInstructions: Write your work up neatly and attach to this page. Record your final answers (only)directly on this page if they are short; if too long indicate which page of the work the answer is on andmark it clearly. Use exact values unless specifically asked to round.1.Find the area bounded by the curves and sketch the region.a. f(x)=x² - 4x² and g(x)=x² - 4xb.y = x² - 2x, y = x +4c. y = cos x, y = 2d. y = x³, y = xe. y = |x|, y = x² - 2cos x, [0, 2π]2. Set up the integral to find the value of the area bounded by the graphs. You do not need toevaluate it, but do sketch the region.a.y =21+x4, y = x²b. y = cos x, y = x + 2 sin4 xi.ii.y-axisthe line y = 83. Calculate the volumes of the solids of revolution as indicated below. Calculate the volume bothwith the shell method and with the disk or washer method and verify that the volume is thesame in each case.a. y=2x², y=0, x=2 Revolved around the:10b. y=₁y=0,₂x = 1, x = 5 Revolved around:i.y-axisy = 10ii.c. y = ex, y = 1, x = 2 Revolved around:i.y-axisx-axisii.d. x = y², x = 1 - y² Revolved around:i.y = 1e. y = x³, y = 0, x = 1 Revolved around:i.ii.y-axisx-axis4. Find the surface area when the function y=x³1.3f. y = ex, y = x² − 1, x = −1, x = 1g. x = 1 - y², x = y² - 1h. x = y¹, y = √2x, y = 01Xi.y ==, y = x, y = x, x > 0j. y = x²e-x, y = xe¯xc. y = e¹-x²1, y=x4+6 2xiii. x-axisiv. the line x = 2iii. x-axisiv. x = 5iii. y = 0iv. x = 3ii. x = 3iii. x = 1iv. y = 1on the interval 1≤x≤2.5. Find the area of the surface generated by revolving the curve about i) the x-axis, ii) the y-axis.Sketch the graph. You may need to integrate numerically.a. x=t, y=4-2t [0,4]d. x = a cos 0, y = asin0 [0,2π]b. x = t³, y = t², [0,1]e. x = t cost, y = t sint, [0,1] C. x = sint, y = sin 2t,f. x = et ―t, y = 4et/2, [0,1]6. A torus is formed by revolving the region bounded by the circle x² + y² = 4 around the linex=3. Find the volume of this doughnut-shaped solid. [Hint: find the function that represents thesemicircle.]7. A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of12-x and the x-axis, around the x-axis, where x and y are measured in meters. Findthe tank's volume.x²818. An ornamental light bulb is designed by resolving the graph of y=x2-x/² on the interval3[0,1/3] around the x-axis, where x and y are measured in feet. Find the surface area of the bulband use the result to approximate the amount of glass needed to make the bulb (assume thatthe glass is 0.015 inches thick).9. Find the volume of the region bounded by y = ex, y = ex, x = 1 around the y-axis. (Useshells.)10. Differentiate.a.[0,1]F(x) = [ - dttb.h(x) = f*t cos(t²) dtSoc. y(x) = ex² e-t² dtd. u(t)= ecos(sin² 3t)e. q(t)cos(Int)tf. v(t) = tan=3a. y = 1 + 6x², [0,1]b. y = (1-e-*), [0,2]x3c. y = ¹+1,[1,2]4xd.3y = x sinx, [0,2π]g. g(x)=√t + sint dt-x eth. r(x) = f == dt√√xti.a(t) = 2t-1j. p(x)k.s(t)11. Find the length of arc in rectangular coordinates. You may need to calculate the valuenumerically.√x² + 3arcsin(et + t)e. y = In(secx), [0,7]3f. y = √√x, [0,1]g. y = √x - x² + arcsin x, [0,1]h. y = e-x²,[1,1]

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