Find the volume of the solid generated by rotating the region bounded by 𝑦 = 𝑒2𝑥, 𝑥-axis, 𝑦-axis and 𝑥 = ln3 about
(i) the 𝑥-axis for 1 complete revolution.
(ii) the 𝑦-axis for 1 complete revolution.
(iii) 𝑦 = −1 for 1 complete revolution.
(iv) 𝑥 = −1 for 1 complete revolution.
as usual, the volumes can be done using shells (v = 2πrh) or discs (washers) (v=πr^2), both multiplied by the thickness (dx or dy). I'll show both integrals
(i)
v = ∫[0,ln3] π(e^(2x))^2 dx = 20π
v = π(1^2)(ln3) + ∫[1,9] 2πy(ln3 - 1/2 lny) dy = ln3 π + 20π - ln3 π = 20π
(ii)
v = π(ln3)^2 + ∫[1,9] π((ln3)^2 - (1/2 lny)^2) dy = (9log3-4)π
v = ∫[0,ln3] 2πxe^(2x) dx = (9log3-4)π
Do the others the same way, but the radius is increased by 1.
To find the volume of the solid generated by rotating a region about an axis, we can use the method of cylindrical shells.
(i) To rotate the region bounded by 𝑦 = 𝑒^2𝑥, 𝑥-axis, 𝑦-axis, and 𝑥 = ln3 about the 𝑥-axis for 1 complete revolution, we can use the following steps:
1. First, find the limits of integration. Since the region is bounded by the 𝑦-axis and the curve 𝑦 = 𝑒^2𝑥, we need to find the intersection points between the curve and the 𝑦-axis.
Setting 𝑦 = 0, we have 0 = 𝑒^2𝑥.
Taking the natural logarithm of both sides, we get ln(0) = ln(𝑒^2𝑥).
Since 𝑒^2𝑥 is always positive, we can disregard the logarithm of 0.
Therefore, there are no intersection points between the curve and the 𝑦-axis.
2. Next, find the height of the cylindrical shell. The height of the shell will be the difference between the values of 𝑦 at each value of 𝑥.
Therefore, the height of the shell is given by 𝑦 = 𝑒^2𝑥.
3. Calculate the circumference of the cylindrical shell. The circumference is given by the formula 2𝜋𝑟, where 𝑟 is the value of 𝑥.
Therefore, the circumference is given by 2𝜋𝑥.
4. Now, we can set up the integral to find the volume of each cylindrical shell. The volume of each shell is given by 𝑉 = 2𝜋𝑥(𝑦) 𝑑𝑥.
Therefore, the integral becomes ∫𝑉 = ∫2𝜋𝑥(𝑒^2𝑥) 𝑑𝑥, with the limits of integration being the points of intersection (which we found to be nonexistent).
5. Evaluate the integral to find the volume.
∫2𝜋𝑥(𝑒^2𝑥) 𝑑𝑥 = 2𝜋 ∫𝑥(𝑒^2𝑥) 𝑑𝑥
To evaluate this integral, we can use integration by parts. Let 𝑢 = 𝑥 and 𝑑𝑣 = 𝑒^2𝑥 𝑑𝑥.
Therefore, 𝑑𝑢 = 𝑑𝑥 and 𝑣 = (1/2)𝑒^2𝑥.
Using the integration by parts formula, the integral becomes:
2𝜋[(𝑥)(1/2)𝑒^2𝑥 - ∫(1/2)𝑒^2𝑥 𝑑𝑥] = 𝜋𝑥𝑒^2𝑥 - ∫(1/2)𝑒^2𝑥 𝑑𝑥
Applying integration by parts again to the remaining integral, we have:
𝜋𝑥𝑒^2𝑥 - [(1/4)𝑒^2𝑥 - (1/4)𝑒^2𝑥 + C] = 𝜋𝑥𝑒^2𝑥 - (1/4)𝑒^2𝑥 + C
6. Finally, evaluate the volume by substituting the limits of integration into the result of the integral.
Since we found that there are no intersection points between the curve and the 𝑦-axis, the volume of the solid generated by rotating the region about the 𝑥-axis for 1 complete revolution is 0.
The same process can be followed for parts (ii), (iii), and (iv) to find the volume of the solid generated by rotating the region about the 𝑦-axis, 𝑦 = -1, and 𝑥 = -1, respectively.