Problem 14 - Calculus: Early Transcendental Functions

Question

Evaluate the line integral. \(\int_{C} 3 y d s,\) where \(C\) is the portion of \(y=x^{2}\) from (0,0) to (2,4)

Step-by-Step Solution

Answer

The solution will vary based on the exact integration and simplification procedure that is chosen. Depending on rounding, the answer will be a positive decimal.

Step 1: Parametrize the Path

Given the function \(y=x^2\), the parameterization of the curve with respect to \(x\) is simply \((x, x^2)\), where \(x\) ranges from 0 to 2. Therefore, we have \(\mathbf{r}(x) = (x, x^2)\) and \(x: [0, 2]\). The derivative of \(\mathbf{r}(x)\) with respect to \(x\) is \(\mathbf{r}'(x) = (1, 2x)\).

Step 2: Compute the magnitude of the derivative

We compute the magnitude of the velocity vector which in this case is \(\sqrt{(dx/dt)^2 + (dy/dt)^2} dt \). We substitute into this the derivative from step 1 and we get \(ds = \sqrt{1 + (2x)^2} dx\). We simplify this to \(ds = \sqrt{1 + 4x^2} dx\).

Step 3: Evaluate the Line Integral

Now we substitute into the line integral, which gives\(\int_0^2 3x^2 \sqrt{1+4x^2} dx\). Then, proceed to find the definite integral. The quartic integral that we end up with here, though it may seem complex and difficult to solve, can be simplified with the substitution \(u = 4x^2 + 1\), which simplifies the integrand substantially. After the integration, simplify the function and then evaluate the definite integral by substituting the upper and lower integration limits.

Step 4: Compute and simplify the result

After finding the antiderivative and substituting in the limits of integration, the value of the line integral can be computed as the difference between the antiderivative evaluated at the upper and lower limits. Evaluate this and simplify.