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Z=1 -x=ln4 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 Ron the xy-plane. R is bounded by the planes y=0, x-In2, x-In4 and the curve C which is parametrised by x= In (t+2) y = (t+1) t>-1 i. Write down an expression for the arc length of C between its intersection points with x-In4 and In2 (blue dots in Structure D). Leave the expression in integral form. ii- Express the equation for the curve C in y=f(x) form.

Z=1 -x=ln4 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 Ron the xy-plane. R is bounded by the planes y=0, x-In2, x-In4 and the curve C which is parametrised by x= In (t+2) y = (t+1) t>-1 i. Write down an expression for the arc length of C between its intersection points with x-In4 and In2 (blue dots in Structure D). Leave the expression in integral form. ii- Express the equation for the curve C in y=f(x) form.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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MAST20029 Engineering Mathematics Formulae Sheet 1. Change of Variable of Integration in 2D JJ₁ f(x, y) dxdy = [[ f(x(u, v), y(u, v))|J(u, v)| dudv R 2. Transformation to Polar Coordinates x = r cos 0, 3. Change of Variable of Integration in 3D [[[ f(x, y, z) dadydz 4. Transformation to Cylindrical Coordinates x = r cos 0, y = r sin 0, 5. Transformation to Spherical Coordinates 7. Work Integrals y = r sin 0, LFG 8. Surface Integrals x = r cos sin o, y = r sin sin o, z = r cos o, = = [[[_ F(u, v), w)|J(u, v, w) dudvdw [[₁, g(x, y, z) ds = S J(r,0) = 6. Line Integrals Jo f(x, y, 2) ds = ["* f(x(t), y(t), z(t)) √x'(t)² + y(t)}² + 2'(t)² dt F(x, y, z). dr = J(r, 0, z) = r = r cb dx - [² R² +F="/ F₁ F₂ dt a F. ÂdS = J(r, 0, 0) = r² sin o 9. Flux Integrals For a surface with upward unit normal, 11. F + F3 dz = [[ g(x, y, f (x, y)) √[f² + f² +1dxdy dt = SS₁₂ - ² -F₁fx - F2fy + F3 dydx
Transcribed Image Text:MAST20029 Engineering Mathematics Formulae Sheet 1. Change of Variable of Integration in 2D JJ₁ f(x, y) dxdy = [[ f(x(u, v), y(u, v))|J(u, v)| dudv R 2. Transformation to Polar Coordinates x = r cos 0, 3. Change of Variable of Integration in 3D [[[ f(x, y, z) dadydz 4. Transformation to Cylindrical Coordinates x = r cos 0, y = r sin 0, 5. Transformation to Spherical Coordinates 7. Work Integrals y = r sin 0, LFG 8. Surface Integrals x = r cos sin o, y = r sin sin o, z = r cos o, = = [[[_ F(u, v), w)|J(u, v, w) dudvdw [[₁, g(x, y, z) ds = S J(r,0) = 6. Line Integrals Jo f(x, y, 2) ds = ["* f(x(t), y(t), z(t)) √x'(t)² + y(t)}² + 2'(t)² dt F(x, y, z). dr = J(r, 0, z) = r = r cb dx - [² R² +F="/ F₁ F₂ dt a F. ÂdS = J(r, 0, 0) = r² sin o 9. Flux Integrals For a surface with upward unit normal, 11. F + F3 dz = [[ g(x, y, f (x, y)) √[f² + f² +1dxdy dt = SS₁₂ - ² -F₁fx - F2fy + F3 dydx
Z=1 Z -x=ln4 V. Find the area of R 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 on the xy-plane. R is bounded by the planes y=0, x=ln2, x-ln4 and the curve C which is parametrised by x= In (t+2) y = (t+1), t>-1 ) STRUCTURE D Curve C x=ln2 i. Write down an expression for the arc length of C between its intersection points with x-ln4 and In2 (blue dots in Structure D). Leave the expression in integral form. ii. Express the equation for the curve C in y=f(x) form. ii. Express the area of R (and thus the volume of D) in terms of integral(s) using: i. Horizontal strip method ii. Vertical strip method dt iv. Show that the area can be expressed in terms of the following single integral Area of R = S²₁ (t+1) (++2) Vi.Evaluate the integral in part (d) using MATLAB.
Transcribed Image Text:Z=1 Z -x=ln4 V. Find the area of R 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 on the xy-plane. R is bounded by the planes y=0, x=ln2, x-ln4 and the curve C which is parametrised by x= In (t+2) y = (t+1), t>-1 ) STRUCTURE D Curve C x=ln2 i. Write down an expression for the arc length of C between its intersection points with x-ln4 and In2 (blue dots in Structure D). Leave the expression in integral form. ii. Express the equation for the curve C in y=f(x) form. ii. Express the area of R (and thus the volume of D) in terms of integral(s) using: i. Horizontal strip method ii. Vertical strip method dt iv. Show that the area can be expressed in terms of the following single integral Area of R = S²₁ (t+1) (++2) Vi.Evaluate the integral in part (d) using MATLAB.
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