Set up an integral for the area of the surface obtained by rotating the given curve about the specified axis. Then evaluate your integral numerically, correct to four decimal places.

Questions:

Set up an integral for the area of the surface obtained by rotating the given curve about the specified axis. Then evaluate your integral numerically, correct to four decimal places.

21. y=e−x2,−1⩽x⩽1;x-axis

22. xy=y2−1,1⩽y⩽3;x-axis

23. x=y+y3,0⩽y⩽1;y-axis

24. y=x+sinx,0⩽x⩽2π/3;y-axis

25. lny=x−y2,1⩽y⩽4;x-axis

26. x=cos2y,0⩽y⩽π/2;y-axis

Answers


Integrals for Surface Area:

Here are the integrals for the area of the surface obtained by rotating the given curves about the specified axes:

21. y=e−x2, −1≤x≤1; x-axis

Surface of Revolution Formula:

S = ∫_a^b 2πy √(1 + (dy/dx)^2) dx

Calculation:

y = e^(-x^2) dy/dx = -2xe^(-x^2)

Integral:

S = ∫_{-1}^1 2πe^(-x^2) √(1 + 4x^2e^(-2x^2)) dx

Note: This integral needs to be solved numerically, unfortunately, there is no analytical solution.

22. xy=y^2−1, 1≤y≤3; x-axis

Surface of Revolution Formula:

S = ∫_a^b 2πx √(1 + (dy/dx)^2) dy

Calculation:

x = (y^2 – 1)/y dy/dx = (2y – 1)/y^2

Integral:

S = ∫_1^3 2π(y^2 – 1)/y √(1 + (2y – 1)^2/y^4) dy

Note: Again, this integral needs to be solved numerically.

23. x=y+y^3, 0≤y≤1; y-axis

Surface of Revolution Formula:

S = ∫_a^b 2πy √(1 + (dx/dy)^2) dy

Calculation:

dx/dy = 1 + 3y^2

Integral:

S = ∫_0^1 2πy √(1 + (1 + 3y^2)^2) dy

Note: This integral can be solved analytically or numerically. Choose your preferred method.

24. y=x+sinx, 0≤x≤2π/3; y-axis

Surface of Revolution Formula:

S = ∫_a^b 2πx √(1 + (dy/dx)^2) dy

Calculation:

dy/dx = 1 + cosx

Integral:

S = ∫_0^(2π/3) 2π(x + sinx) √(1 + (1 + cosx)^2) dx

Note: This integral needs to be solved numerically.

25. lny=x−y2, 1≤y≤4; x-axis

Surface of Revolution Formula:

S = ∫_a^b 2πy √(1 + (dy/dx)^2) dx

Calculation:

x = ln(y) + y^2 dy/dx = 1/y + 2y

Integral:

S = ∫_1^4 2π(ln(y) + y^2) √(1 + (1/y + 2y)^2) dx

Note: This integral needs to be solved numerically.

26. x=cos^2y, 0≤y≤π/2; y-axis

Surface of Revolution Formula:

S = ∫_a^b 2πx √(1 + (dx/dy)^2) dy

Calculation:

dx/dy = -2cos(y)sin(y)

Integral:

S = ∫_0^(π/2) 2πcos^2(y) √(1 + (2cos(y)sin(y))^2) dy

Note: This integral can be solved analytically or numerically. Choose your preferred method.

Numerical Evaluation:

Once you have the integral, you can use a numerical integration tool or software to evaluate it and obtain the surface area to four decimal places. Remember to specify the integration limits and initial conditions accurately while using the software.

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