STEM-Master
Calculus IIIChapter 5: Line IntegralsSection 5-5

Fundamental Theorem for Line Integrals — Question 6

PDF ↗

Question 6

Let f(x,y)=x2+y2f(x,y)=x^2+y^2. A curve begins at A=(1,2)A=(1,2) and ends at a first-quadrant point BB on the line y=x+1y=x+1. Suppose ∫C∇f⋅d𝒓=20.\int_C\nabla f\cdot d\mathbf r=20. Determine BB.

Tasks

  1. Convert the work condition into a level-curve condition on BB.

  2. Solve the line-circle intersection equation.

  3. Use the quadrant restriction to select and verify the endpoint.

Original worksheet page 1: question and worked solution for 5-5-006
Show solutionHide solution

Question 6 – Solution

Strategy. Use the theorem in reverse: the known integral determines the terminal potential value.

Step 1: Determine the level Since f(A)=12+22=5f(A)=1^2+2^2=5, 20=f(B)−5,f(B)=25.20=f(B)-5, \qquad f(B)=25. Thus BB lies on x2+y2=25x^2+y^2=25 as well as y=x+1y=x+1.

See the diagram in the original worksheet below.

Step 2: Solve the intersection Substitute y=x+1y=x+1: x2+(x+1)2=25⇒2x2+2x−24=0⇒(x−3)(x+4)=0.x^2+(x+1)^2=25 \Longrightarrow 2x^2+2x-24=0 \Longrightarrow (x-3)(x+4)=0. The candidates are (3,4)(3,4) and (−4,−3)(-4,-3).

Step 3: Apply the restriction Only (3,4)(3,4) lies in the first quadrant, so B=(3,4).\boxed{B=(3,4)}.

Verification The line condition gives 4=3+14=3+1, and f(B)−f(A)=25−5=20f(B)-f(A)=25-5=20, exactly the specified integral.

Original worksheet page 2: question and worked solution for 5-5-006

Original worksheet layout. Use Enlarge or open the PDF for a closer view.

← Section questionsChapter contents