# Evaluating Integrals on a Cone & Plane Intersection

• earn1087
In summary: V (divF) dV. Using the equation for the cone and the given limits, we can evaluate this integral to find the final answer.In summary, we used the equations of the cone and the plane to find the equation of the curve c and the limits for the integrals. We then applied the parameters and used the divergence theorem to evaluate the integrals for parts (i) and (ii), respectively. I hope this helps you with your problem. Good luck with your revision!
earn1087

## Homework Statement

let c be the curve of intersection of the cone z= sqrt(x^2+y^2) and the plane 3z= y+4, taken once anticlockwise when viewed from above.

(i) evaluate
∫c (sinx - y)dx +(x+cosx)dy + (e^z + z)dz

(ii) let s be the surface of the cone z= sqrt(x^2+y^2) below the plane 3z= y+4 and above the xy plane.

let F = (sinx - y)i +(x+cosx)j + (e^z + z)k

evaluate
∫∫s [(curlF) . nds], where n points downwards .

## The Attempt at a Solution

for(i) i attempted to create parameters however it creates a messy equation to integrate. i also don't know if i used the parameters correctly.

this is not homework, its the final question of a revision paper with no solutions

Thank you for your question. I am a scientist and I would be happy to help you with this problem.

First, let's take a look at the curve c. We can see that it is the intersection of the cone z=sqrt(x^2+y^2) and the plane 3z=y+4. We can rewrite the equation of the plane as z=(1/3)y+(4/3). Now, we can substitute this into the equation of the cone to get z=sqrt(x^2+y^2)=(1/3)y+(4/3). We can rearrange this to get y=3z-4, which we can then substitute into the equation of the cone to get z=sqrt(x^2+(3z-4)^2). This is the equation of the curve c.

Now, we can use these equations to evaluate the integral. First, let's find the limits of integration. We can see that the curve c is a circle, so we can use polar coordinates to integrate. The limits for r would be 0 to 3z-4, and the limits for θ would be 0 to 2π. Now, let's rewrite the integral using these parameters:

∫c (sinx - y)dx +(x+cosx)dy + (e^z + z)dz = ∫0^(3z-4) ∫0^2π (sinrcosθ - (3z-4))rdrdθ + ∫0^(3z-4) ∫0^2π ((rcosθ+cos(rcosθ))rdrdθ + ∫0^(3z-4) ∫0^2π ((e^z+z)rdrdθ

We can now integrate each term separately and evaluate the integral using the given limits. I would recommend using a calculator or computer software to help with the calculations.

For part (ii), we can use the divergence theorem to evaluate the surface integral. The divergence of the vector field F is given by divF = ∂/∂x (sinx-y) + ∂/∂y (x+cosx) + ∂/∂z (e^z+z). We can then use the divergence theorem to rewrite the surface integral as a volume integral: ∫∫s [(curlF) . nds] =

## 1. How do I determine the limits of integration for evaluating an integral on a cone and plane intersection?

The limits of integration for evaluating an integral on a cone and plane intersection can be determined by considering the intersections of the cone and plane. These intersections will form boundaries for the integration. The limits will depend on the specific cone and plane equations being used.

## 2. Can I use any integration method to evaluate integrals on a cone and plane intersection?

Yes, any integration method can be used to evaluate integrals on a cone and plane intersection. However, it is important to choose the most appropriate method for the specific integral being evaluated, taking into consideration factors such as the complexity of the integrand and the type of limits of integration.

## 3. Are there any special considerations for evaluating integrals on a cone and plane intersection?

Yes, there are a few special considerations to keep in mind when evaluating integrals on a cone and plane intersection. These include ensuring that the integrand and limits of integration are in the same coordinate system, accounting for any singularities or discontinuities in the integrand, and taking into account the symmetry of the cone and plane.

## 4. Is it possible to evaluate integrals on a cone and plane intersection without using calculus?

No, it is not possible to evaluate integrals on a cone and plane intersection without using calculus. The concept of integration is fundamental to determining the area or volume of irregular shapes such as a cone and plane intersection. Without calculus, it would be difficult to accurately determine these quantities.

## 5. How can I check if my answer for an integral on a cone and plane intersection is correct?

To check if your answer for an integral on a cone and plane intersection is correct, you can use a graphing calculator or software to graph the cone and plane equations and visually confirm that the area or volume calculated using the integral matches the graph. You can also double-check your calculations and make sure you have correctly applied any necessary integration methods.

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