Verify Stokes’s Theorem by evaluating $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ as a line integral and as a double integral. $\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}$, $S: 6 x+6 y+z=12$, first octant

Verify Stokes’s Theorem by evaluating ∫_c F·T ds = ∫_c F·dr as a line integral and as a double integral. F(x, y, z) = xyzi + yj + zk S: 6x + 6y + z = 12, first octant

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The sphere r(u, v) = a sin u cos vi + a sin u sin vj + a cos uk, where 0 ≤ u ≤ π and 0 ≤ v ≤ 2π

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The surface of revolution r(u, v) = √u cos vi + √u sin vj + uk, where 0 ≤ u ≤ 4 and 0 ≤ v ≤ 2π

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the paraboloid r(u, v) = 2u cos vi + 2u sin vj + u²k, where 0 ≤ u ≤ 2 and 0 ≤ v ≤ 2π

What is a doubling time? Suppose a population has a doubling time of 25 years. By what factor will it grow in 25 years? in 50 years? in 100 years?

In the following exercise, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point.

$$r(u, v)=2 u \cosh v i+2 u \sinh v j+\frac{1}{2} u^2 k,(-4,0,2)$$

Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. r(u, v) = 2u cos vi + 3u sin vj + u²k, (0, 6, 4)

Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. r(u, v) = ui + vj + √uv k, (1, 1, 1)

Use Green’s Theorem to evaluate the line integral.

$$∫_c e^x cos 2y dx - 2e^x sin 2y dy C: x² + y² = a²$$

Algebraically determine the general solution to the equation $4 \cos ^{2} x-1=0$. Give your answer in radians.

Use Green’s Theorem to evaluate the line integral. ∫_c 2xy dx + (x+y) dy C: boundary of the region lying between the graphs of y = 0 and y = 1 - x²

Use Green’s Theorem to evaluate the integral ∫_c (y-x) dx + (2x - y) dy for the given path. C: boundary of the region lying between the graphs of y = x and y = x² - 2x

Evaluate

$$f_CF\cdot dr$$

along the curve C. F(x, y)=x²i+xyj, C:r(t0=2costi+2sintj (0≤t≤π)