Problem 1
Let , and . Denote by the part of the surface with , .
(a). Find the Surface Area of without Using Any Calculus
Problem 2
Let . Denote by the triangle with vertices , , and .
(b). Denote by the Projection of onto the -plane (it is the Triangle with Vertices , , and ). Similarly Use to Denote the Projection of onto the -plane and to Denote the Projection of onto the -plane. Show that $$\text{Area}(S) = \sqrt{\text{Area}(T_{xy})^2 + \text{Area}(T_{xz})^2 + \text{Area}(T_{yz})^2}.$$
Problem 3
Let . Denote by the part of the cylinder with , , and .
(a). Find the Surface Area of without Using Any Calculus
(b). Parametrise by $$\vec{r}(\theta, y) = a\cos\theta,\hat{i} + y,\hat{j} + a\sin\theta,\hat{k}, \qquad 0 \le \theta \le \frac{\pi}{2},\ 0 \le Y \le h,$$
and find the surface area of by using the surface area formula for parametrised surfaces
Problem 4
Let be the part of the surface lying inside the cylinder . Find the moment of inertia of about the -axis, that is,
Problem 5
Find the surface area of the part of the paraboloid which lies above the -plane.
Problem 6
Find the area of the portion of the cone lying between the planes and .
Problem 7
Determine the surface area of the surface given by , over the square , .
Problem 8
(a). To Find the Surface Area of the Surface above the Region , We Integrate . what is ?
(b). Consider a "Death Star": a Ball of Radius Centred at the Origin with Another Ball of Radius Centred at Cut out of It. the Diagram Shows the Slice where
(i). The Rebels want to Paint part of the Surface of the Death Star Hot Pink; Specifically, the Concave part (indicated with a Thick line in the diagram). to Help Them Determine how much Paint is Needed, Carefully Fill in the Missing Parts of This Integral: $$\text{surface area} = \int_{\underline{\ \ \ \ }}^{\underline{\ \ \ \ }}\int_{\underline{\ \ \ \ }}^{\underline{\ \ \ \ }}\underline{\ \ \ \ \ \ \ \ \ \ },dr,d\theta.$$
(ii). What is the Total Surface Area of the Death Star?
Problem 9
Find the area of the portion of the cone lying in the first octant between the planes and .
Problem 10
Find the surface area of that part of the hemisphere which lies within the cylinder .
Problem 11
The cylinder cuts out a portion of the upper half of the cone . Compute
Problem 12
Find the surface area of the torus obtained by rotating the circle (the circle is contained in the -plane) about the -axis.
Problem 13
A spherical shell of radius is centred at the origin. Find the centroid (i.e. the centre of mass with constant density) of the part of the sphere that lies in the first octant.
Problem 14
Find the area of that part of the cylinder lying outside .
Problem 15
Let and be positive constants, and let be the part of the conical surface
where . Consider the surface integral .
(a). Express as a Double Integral over a Disk in the -plane
(b). Use the Parametrisation , , Etc., to Express as a Double Integral over a Suitable Region in the -plane
(c). Evaluate Using the Method of Your Choice
Problem 16
Evaluate, for each of the following, the flux where is the outward normal to the surface .
(a). And the Surface is the Sphere
(b). And is the Surface of the Rectangular Box , ,
(c). And is the Surface of the Solid Cone
Problem 17
Let be the part of the surface that lies above the square , .
(a). Find
(b). Find the Flux of upward through
Problem 18
Let be the part of the surface that lies above the square , in the -plane.
(a). Find
(b). Find the Flux of upward through
Problem 19
Find the area of the part of the surface that lies above .
Problem 20
Let be the spherical cap which consists of the part of the sphere which lies under the plane . Let . Calculate
Problem 21
(a). Find a Parametrization of the Surface of the Cone whose Vertex is at the point , and whose Base is the Circle in the -plane. only the Cone Surface Belongs to , not the Base. Be Careful to Include the Domain for the Parameters
(b). Find the -coordinate of the Centre of Mass of the Surface from (a)
Problem 22
Let be the surface of a cone of height and base radius . The surface does not include the base of the cone or the interior of the cone. Find the centre of mass of .
Locate the cone in a coordinate system so that its base is in the -plane, and its vertex on the -axis, at the point . The base is a circle of radius in the -plane with centre at the origin. The cone surface is characterized by the fact that for every point of , the distance from the -axis and the distance from the -plane add up to .
Problem 23
Let be the portion of the elliptical cylinder lying between the planes and , and let denote the outward normal to . Let . Calculate the flux integral directly, using an appropriate parametrization of .
Problem 24
Evaluate the flux integral where , and is the part of the paraboloid that lies above the triangle , . is oriented so that its unit normal has a negative -component.
Problem 25
Evaluate the surface integral where is the part of the sphere for which .
Problem 26
Let be the surface given by the equation
lying between the planes and . Evaluate the integral
Problem 27
Let be the part of the paraboloid lying above the -plane. At , has density
Find the centre of mass of .
Problem 28
Let be the part of the plane that lies in the first octant, oriented so that has a positive component. Let . Evaluate the flux integral .
Problem 29
Find the net flux of the vector field upwards (with respect to the -axis) through the surface parametrised by for , .
Problem 30
Let be the surface obtained by revolving the curve , , around the -axis, with the orientation of having pointing toward the -axis.
(a). Draw a Picture of and Find a Parametrisation of
(b). Compute the Integral
(c). Compute the Flux Integral where
Problem 31
Compute the net outward flux of the vector field
across the boundary of the region between the spheres of radius and radius centred at the origin.
Problem 32
Evaluate the surface integral where is the part of the cone where and .
Problem 33
Compute the flux integral , where
and is the part of the paraboloid lying inside the cylinder , with orientation pointing downwards.
Problem 34
Let the thin shell consist of the part of the surface with , , and . Find the mass of if it has surface density given by kg per unit area.
Problem 35
Let be the portion of the paraboloid that satisfies . Its unit normal vector is chosen so that . Find the flux of out of .
Problem 36
Let denote the portion of the paraboloid for which . Orient so that its unit normal has a positive component. Let
Evaluate the surface integral .
Problem 37
Let be the boundary of the apple core bounded by the sphere and the hyperboloid . Find the flux integral where and is the outward normal to the surface .
Application
Problem 38
(a). Consider the Surface given by the Equation $$x^2 + z^2 = \cos^2 y.$$
Find an equation for the tangent plane to at the point
(b). Compute the Integral $$\iint_S \sin y,dS$$
where is the part of the surface from (a) lying between the planes and
Problem 39
Let be a function on such that all its first-order partial derivatives are continuous. Let be the surface for some . Assume that on . Let be the gradient field .
(a). Let Be a Piecewise Smooth Curve Contained in (not Necessarily closed). Must it Be True that ? Explain why
(b). Prove that for Any Vector Field , $$\iint_S (\vec{F}\times\vec{G})\cdot\hat{n},dS = 0.$$
Problem 40
(a). Give Parametric Descriptions of the Form for the following Surfaces. Be Sure to State the Domains of Your Parameterizations
(i). The part of the Plane in the First Octant , ,
(ii). The Cap of the Sphere for
(iii). The Hyperboloid for
(b). Use Your Parametrization from part (a) to Compute the Surface Area of the Cap of the Sphere for
Problem 41
Let be the part of the sphere where , oriented away from the origin.
(a). Compute
(b). Compute
Problem 42
Let be the part of the surface which lies in the first octant. Find the flux of downwards through where