Questions: Evaluate [ intc 4 x^2 d x ] c is a line segment joining (0,2) to (2,0)

Solution Steps

1. Parameterize the line segment joining (0,2) to (2,0).
2. Express \( x \) and \( y \) in terms of a parameter \( t \) that varies from 0 to 1.
3. Substitute these parameterized expressions into the integral.
4. Evaluate the resulting integral with respect to \( t \).

The line segment joining the points \((0, 2)\) and \((2, 0)\) can be parameterized using the variable \(t\) as follows: \[ x = 2t, \quad y = 2 - 2t \quad \text{for } t \in [0, 1]. \]

To evaluate the integral, we need to compute the derivative of \(x\) with respect to \(t\): \[ \frac{dx}{dt} = 2. \]

Substituting the parameterization into the integrand \(4x^2 \, dx\), we have: \[ \text{Integrand} = 4(2t)^2 \cdot 2 = 32t^2. \]

Now, we evaluate the integral: \[ \int_{0}^{1} 32t^2 \, dt. \] Calculating this gives: \[ \int_{0}^{1} 32t^2 \, dt = \left[ \frac{32}{3} t^3 \right]_{0}^{1} = \frac{32}{3}. \]

Final Answer