Sir i need help in two questions, i am stuck !
1) Calculate the population within a 7mile radius if the radialpopulation density is 6(3+r^2)^1/3 ? Use decimal notation and aswer should be in million.
2) Find the volume of solid obtained by rotating the region enclosed by y=x^2 and y=5x about the line x=0. kindly tell me the upper limit and integrand of it, i will solve then by myself.
The population in a ring of radius r is density * area, so adding up all the rings you get
∫[0,7] 6∛(3+r^2) * 2πr dr
hint: let u = 3+r^2
for the volume, you can use
shells:
∫[0,5] 2πrh dx
= ∫[0,5] 2πx(5x-x^2) dx
discs:
∫[0,25] π(R^2-r^2) dy
= ∫[0,25] π(y - (y/5)^2) dy
done thank you steve
Certainly! I'd be happy to help you with your questions.
1) To calculate the population within a 7-mile radius, we need to find the area of the circle with a radius of 7 miles and multiply it by the radial population density. Here's how you can do it:
Step 1: Find the area of the circle
The formula for the area of a circle is A = πr^2, where "r" is the radius. In this case, the radius is 7 miles. So, the area can be calculated as:
A = π(7)^2 = 49π square miles.
Step 2: Multiply the area by the radial population density
The radial population density function is given as 6(3 + r^2)^1/3. Since we're dealing with a circle with a constant radius of 7 miles, we can substitute "r" with 7 in the formula to find the population within the circle:
Population = 6(3 + 7^2)^1/3.
Now, you can evaluate this expression to find the population within the 7-mile radius. Remember to use decimal notation and express the answer in million.
2) To find the volume of the solid obtained by rotating the region enclosed by y = x^2 and y = 5x about the line x = 0, you can use the method of cylindrical shells. Here's how you can find the upper limit and integrand:
Step 1: Determine the limits of integration
To find the limits of integration, you need to determine the x-values where the two curves intersect. In this case, the curves y = x^2 and y = 5x intersect at x = 0 and x = 5. Therefore, the upper limit of integration would be x = 5.
Step 2: Determine the integrand
The integrand is the function that you will integrate with respect to x. In the case of cylindrical shells, the height of each shell will be the difference between the two curves, and the radius will be the distance from the axis of rotation (in this case, x = 0) to the curve y = x^2.
For this problem, the radius will be x, and the height will be the difference between the two curves: (5x - x^2).
Therefore, the integrand for this problem will be: 2πx (5x - x^2).
Now, you can set up the integral and solve it using the upper limit of integration and the integrand.
I hope these explanations help you understand how to approach these problems. If you have any further questions, feel free to ask!