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X Z=1 R N -x=ln4 Y Curve C x=In2 Consider a structure D with a curved solar panel wall that is moulded to a curve C, as shown in Figure 2. D occupies the region below the surface z = f(x,y) = 1 and above the region R on the xy-plane. R is bounded by the planes y=0, x=In2, x=In4 and the curve C which is parametrised by x = ln(t + 2), y = (t+1), t>-1 (a) Write down an expression for the arc length of C between its intersection points with x=In4 and In2 (blue dots in Figure 2). Leave the expression in integral form. (b) Express the equation for the curve C in y=f(x) form.

X Z=1 R N -x=ln4 Y Curve C x=In2 Consider a structure D with a curved solar panel wall that is moulded to a curve C, as shown in Figure 2. D occupies the region below the surface z = f(x,y) = 1 and above the region R on the xy-plane. R is bounded by the planes y=0, x=In2, x=In4 and the curve C which is parametrised by x = ln(t + 2), y = (t+1), t>-1 (a) Write down an expression for the arc length of C between its intersection points with x=In4 and In2 (blue dots in Figure 2). Leave the expression in integral form. (b) Express the equation for the curve C in y=f(x) form.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.3: Volume And Average Value
Problem 12E
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X Z=1 R N -x=ln4 Y Curve C x=ln2 Consider a structure D with a curved solar panel wall that is moulded to a curve C, as shown in Figure 2. D occupies the region below the surface z = f(x, y) = 1 and above the region R on the xy-plane. R is bounded by the planes y=0, x=In2, x=In4 and the curve C which is parametrised by x = ln(t + 2), y = (+1)' t>-1 (a) Write down an expression for the arc length of C between its intersection points with x=ln4 and In2 (blue dots in Figure 2). Leave the expression in integral form. (b) Express the equation for the curve C in y=f(x) form.
Transcribed Image Text:X Z=1 R N -x=ln4 Y Curve C x=ln2 Consider a structure D with a curved solar panel wall that is moulded to a curve C, as shown in Figure 2. D occupies the region below the surface z = f(x, y) = 1 and above the region R on the xy-plane. R is bounded by the planes y=0, x=In2, x=In4 and the curve C which is parametrised by x = ln(t + 2), y = (+1)' t>-1 (a) Write down an expression for the arc length of C between its intersection points with x=ln4 and In2 (blue dots in Figure 2). Leave the expression in integral form. (b) Express the equation for the curve C in y=f(x) form.
1. Change of Variable of Integration in 2D [ 1(x, y) dxdy = f(x(u, v),y(u, v))|J(u, v)| dudv 2. Transformation to Polar Coordinates 3. Change of Variable of Integration in 3D JJJ. [ f(x,y, 2) dxdydz = [[[_ F(u,v, w)|J(u, v, w)| dudvdw x=r cos 0, y =rsin 0, J(,0)=r 4. Transformation to Cylindrical Coordinates 5. Transformation to Spherical Coordinates 6. Line Integrals x = r cos 0, y = r sin 0, z = 2, J(r,0,2)=r x = r cos@sind, y=rsin sind, z=rcos, J(r,0,0) = r² sin 7. Work Integrals [ f(x, y, z) ds = [* f(x(1), y(t), =(0)) √/x'(0}² +y°(1)² +2{t}°dt dx dz dt [F(x, y, z) - dr = -C = [° F + F + F = d 8. Surface Integrals [[ 91,9, 2) ds = [[ 9(2.v. f(x,y) √/ S3 + f2 + 1 dxdy R 9. Flux Integrals For a surface with upward unit normal, = 11₁₂₁-B₁ II.F. FindS -Fifz-F₂fy + F3 dydx
Transcribed Image Text:1. Change of Variable of Integration in 2D [ 1(x, y) dxdy = f(x(u, v),y(u, v))|J(u, v)| dudv 2. Transformation to Polar Coordinates 3. Change of Variable of Integration in 3D JJJ. [ f(x,y, 2) dxdydz = [[[_ F(u,v, w)|J(u, v, w)| dudvdw x=r cos 0, y =rsin 0, J(,0)=r 4. Transformation to Cylindrical Coordinates 5. Transformation to Spherical Coordinates 6. Line Integrals x = r cos 0, y = r sin 0, z = 2, J(r,0,2)=r x = r cos@sind, y=rsin sind, z=rcos, J(r,0,0) = r² sin 7. Work Integrals [ f(x, y, z) ds = [* f(x(1), y(t), =(0)) √/x'(0}² +y°(1)² +2{t}°dt dx dz dt [F(x, y, z) - dr = -C = [° F + F + F = d 8. Surface Integrals [[ 91,9, 2) ds = [[ 9(2.v. f(x,y) √/ S3 + f2 + 1 dxdy R 9. Flux Integrals For a surface with upward unit normal, = 11₁₂₁-B₁ II.F. FindS -Fifz-F₂fy + F3 dydx
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