Line Integrals of Vector Fields — Question 1

Question 1

Let $$\mathbf F(x,y)=\langle y,x^2\rangle,\qquad \mathbf r(t)=\langle t,t^2\rangle,\quad 0\le t\le 1.$$ Evaluate the line integral $\displaystyle\int_C\mathbf F\cdot d\mathbf r$.

Tasks

1. Evaluate $\mathbf F$ along the curve and compute $\mathbf r'(t)$.

2. Form the dot product and evaluate the integral.

3. State how the sign of the integrand agrees with the geometry.

Question 1 – Solution

Strategy. Use $\int_C\mathbf F\cdot d\mathbf r=\int_a^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt$.

Step 1: Restrict the field to the curve Since $x=t$ and $y=t^2$, $$\mathbf F(\mathbf r(t))=\langle t^2,t^2\rangle,\qquad \mathbf r'(t)=\langle 1,2t\rangle.$$

Step 2: Form the scalar integrand $$\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)=t^2+2t^3.$$ Therefore $$\int_C\mathbf F\cdot d\mathbf r =\int_0^1(t^2+2t^3)\,dt =\left[\frac{t^3}{3}+\frac{t^4}{2}\right]_0^1 =\boxed{\frac 56}.$$

Verification For $0\le t\le 1$, both components of the field and both components of the velocity are nonnegative. Their dot product must be nonnegative, consistent with the positive answer.