Question

Compute the following integrals: a. \(\int_{C}\left(x^{2}+y\right) d x+\left(3 x+y^{3}\right) d y\) for \(C\) the ellipse \(x^{2}+4 y^{2}=4\). b. \(\int_{S}(x-y) d y d z+\left(y^{2}+z^{2}\right) d z d x+\left(y-x^{2}\right) d x d y\) for \(S\) the positively oriented unit sphere. c. \(\int_{C}(y-z) d x+(3 x+z) d y+(x+2 y) d z\), where \(C\) is the curve of intersection between \(z=4-x^{2}-y^{2}\) and the plane \(x+y+z=0\). d. \(\int_{C} x^{2} y d x-x y^{2} d y\) for \(C\) a circle of radius 2 centered about the origin. e. \(\int_{S} x^{2} y d y d z+3 y^{2} d z d x-2 x z^{2} d x d y\), where \(S\) is the surface of the cube \([-1,1] \times[-1,1] \times[-1,1]\).

Short answer

The solutions to the integrals are obtained by using a combination of parametrization, transformations to polar coordinates, and the Stokes', Green's, and the divergence theorems.

Step-by-step solution

Problem a

To solve \(\int_{C}\left(x^{2}+y\right) d x+\left(3 x+y^{3}\right) d y\), where \(C\) the ellipse \(x^{2}+4 y^{2}=4\), the first step is to parametrize the ellipse \(C\) as \(C: x = 2\cos t, y = \sin t,\) where \(0\leq t\leq 2\pi\). Replace \(x\) and \(y\) in the integral with their respective parametric equations, and evaluate the resulting integral over the interval \(0\leq t\leq 2\pi\).

Problem b

The first step in solving \(\int_{S}(x-y) d y d z+\left(y^{2}+z^{2}\right) d z dx+\left(y-x^{2}\right) d x d y\) is to recognize that the surface integral is over a sphere. Rewrite the integrand by shifting terms to form an exact differential \(dF\). This suggests the use of the divergence theorem for a volume integral instead of a surface one. The divergence theorem implies that you should evaluate the integral \(\iiint_V \nabla \cdot \vec{F} dV \) over the volume of a unit sphere, where the problem is simpler.

Problem c

Here, you need to compute \(\int_{C}(y-z) d x+(3 x+z) d y+(x+2 y) d z\), where \(C\) is the curve of intersection between \(z = 4-x^{2}-y^{2}\) and the plane \(x+y+z = 0\). Write the integral in the form \(\int_C \vec{F} \cdot d\vec{r}\), and use Stokes' Theorem to convert the line integral to a surface integral. Integrate over the closed surface between the two planes. Remember to compute the curl of the vector field \( \vec{F} \) before doing the integration.

Problem d

The problem is to evaluate \(\int_{C} x^{2} y d x-x y^{2} d y\), where \(C\) is a circle of radius 2 centered about the origin. First, you convert the Cartesian coordinate system to polar coordinates: \(x = r\cos(\theta), y = r\sin(\theta)\). The limits of integration in polar coordinates are \(0 \leq r \leq 2\) and \(0 \leq \theta \leq 2\pi\). Then, compute the integral over the disk of radius 2.

Problem e

To evaluate \(\int_{S} x^{2} y d y d z+3 y^{2} d z d x-2 x z^{2} d x d y\), \(\) where \(S\) is the surface of the cube \([-1,1] \times[-1,1] \times[-1,1]\), rewrite the integral as \(\int_S \vec{F} \cdot d\vec{S}\). Use the divergence theorem to convert this surface integral into a volume integral \(\iiint_V div \vec{F} \,dV\) over the volume of the cube. With this reduction, the integral becomes easier to compute.

Key concepts

Surface Integrals

Surface integrals allow us to sum up values over a surface, such as the amount of fluid flowing across a membrane. To compute a surface integral, you need to parameterize the surface, typically in terms of two variables, and integrate the function of interest over this surface. For instance, problem b involves integrating over the sphere's surface, a classic example of a surface integral. It's important to properly orient the surface and understand how the differential elements change when converting from Cartesian coordinates to another system, such as spherical coordinates.

Imagine painting each tiny patch of the surface with the value of the function and then summing up the 'value' painted on the whole surface; that sum is the surface integral. The key to evaluating surface integrals is considering how small changes on the surface relate to the coordinates used in the parameterization.

Line Integrals

Conversely, line integrals, like those in problem a, involve integrating a function along a curve. These are useful for summing values along a path, such as calculating the work done by a force field along a trajectory. For this, we often use parametric equations to express the curve. Understanding the curve's geometry and how to parameterize it is crucial for solving line integrals. A good parameterization will simplify the computation. Frequently, we convert complicated paths into simpler forms through various coordinate systems to easily evaluate these integrals. Think of it like collecting coins along a pathway; the line integral adds up all the coins' values along the path taken.

Stokes' Theorem

Stokes' Theorem bridges the gap between line integrals and surface integrals. It tells us that the integration of certain vector fields around a closed loop can be related to the integration of a vector field over the surface enclosed by the loop (as illustrated in problem c). This theorem is a powerful tool in physics and engineering for simplifying complex line integrals in electromagnetism and fluid dynamics. Using Stokes' Theorem effectively requires a strong grasp of the curl operator, the orientation of surfaces, and the ability to judge when the theorem is applicable to simplify a problem.

Divergence Theorem

The divergence theorem, applied in problem b and problem e, is a key tool for converting surface integrals into volume integrals. The theorem relates the outward flux of a vector field through a closed surface to the integral of the divergence of this vector field over the volume enclosed by the surface. It's instrumental in fields such as fluid dynamics and electromagnetism, simplifying many complex three-dimensional integrals into more manageable ones. When evaluating integrals using this theorem, a comprehensive understanding of divergence and proper setup and execution of triple integrals are imperative for success.

Imagine a balloon expanding within a box where the air inside represents the vector field; the divergence theorem would relate the amount of air pushing outward on the box's surface to the total amount of air within the balloon.

Parametric Equations

Parametric equations are a way of defining a geometric object using parameters. In problems a and d, curves are parameterized to simplify the computation of line integrals. Parametric equations are versatile tools used to trace out paths in multidimensional space, a fundamental step in evaluating both line and surface integrals. When working with these equations, it's vital to have a clear picture of what shape or curve they describe and how they relate to the integration limits.

Exact Differential

An exact differential is a term encountered when a multivariable function has a differential form that can be integrated without reference to a path or surface, as seen in part of problem b. It represents the total differential of a scalar function, a concept that ties closely with line and surface integrals. Recognizing and working with exact differentials enable us to make use of functions' properties to simplify complex integral calculations. It is akin to having a map where the differential gives you reliable directions, hence ensuring that you can travel from point A to point B without worrying about the specific path taken.

Polar Coordinates

Polar coordinates are used when problems exhibit radial symmetry, as in problem d. This two-dimensional coordinate system represents points by how far away and at what angle they are from a reference point. It is a powerful tool to simplify integration via substitution when dealing with circular or spherical symmetry. The circle's nature means that polar coordinates create more straightforward integrals to perform, greatly simplifying problems with circular domains. Converting from Cartesian to polar coordinates involves recognizing the relationships between the two systems, such as how the Cartesian coordinates x and y correspond to the polar coordinates r (radius) and θ (angle).