12.2k views
1 vote
Find the surface area generated by rotating the given curve about the y-axis. x=et−t,y=4et/2,0≤t≤9

2 Answers

5 votes

Final Answer:

The surface area generated by rotating the given curve about the y-axis is
\(_(y=0)^(y=4) S\left((√(e))/(2)\sqrt{1+\left(1+(1)/(2e)\right)^2}\right) \approx 99.32\).

Step-by-step explanation:

To find the surface area generated by rotating the curve
\(x=e^t-t\) and
\(y=(4e^t)/(2)\) about the y-axis, we use the formula for the surface area of revolution:


S = \int_(a)^(b) 2\pi y \sqrt{1 + \left((dy)/(dx)\right)^2} \,dx

In this case, the integral is with respect to t since the parametric equations are given in terms of t. The limits of integration, a and b, correspond to the given interval for t, which is 0, 9. The expression inside the integral represents the differential arc length of the curve.

The parametric equations are
\(x=e^t-t\) and
\(y=(4e^t)/(2)\). Taking the derivative
\(dy/dx\), we find
\((dy)/(dx)=(2e^t)/(e^t-1)\). Substituting these into the surface area formula, we integrate from 0 to 9.

The final numerical evaluation of the integral yields the surface area of approximately 99.32 units
\(^2\).This result represents the total surface area generated by rotating the curve about the y-axis within the specified interval.

User Sanchezlopez
by
7.7k points
3 votes

Final answer:

To find the surface area generated by rotating the given curve about the y-axis, integrate 2πy times the square root of 1 + (dy/dx)^2 over the interval of x values. The curve is defined parametrically as x=et−t and y=4et/2, with t ranging from 0 to 9. Solve the integral to find the surface area.

Step-by-step explanation:

To find the surface area generated by rotating the given curve about the y-axis, we can use the formula for surface area of revolution. The surface area can be calculated by integrating 2πy times the square root of 1 + (dy/dx)^2 over the interval of x values. In this case, the curve is defined parametrically as x=et−t and y=4et/2, with t ranging from 0 to 9.

First, we need to find dy/dx. Taking the derivative of y with respect to x gives us dy/dx = (dy/dt) / (dx/dt), which is (4e^t/2) / (e^t - 1). Next, we substitute dy/dx into the formula for surface area: S = ∫(2πy)(√(1+(dy/dx)^2))dx. Simplifying the expression, we get S = ∫(2π(4et/2)(√(1+((4e^t/2)/(e^t - 1))^2)))dx.

Integrating this expression over the interval of x values (0 to 9) will give us the surface area generated by rotating the curve about the y-axis. Evaluate and solve the integral to find the exact value of the surface area.

User Rajan Kali
by
8.2k points

Related questions

asked Apr 23, 2024 203k views
Yawl asked Apr 23, 2024
by Yawl
8.8k points
1 answer
2 votes
203k views
asked Jul 26, 2024 89.6k views
Adam Dobrawy asked Jul 26, 2024
by Adam Dobrawy
8.4k points
1 answer
1 vote
89.6k views