Fundamental Theorem of Line Integrals - AP Calculus BC

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Question

Evaluate $\int_{C} abla f \cdot d\vec{r}$ , where $f(x,y,z)=\cos(2\pi x)+\sin(3\pi y)+xyz$ , and is any path that starts at , and ends at .

Answer

Since there isn't a specific path we need to take, we just evaluate at the end points.

$\int_{C} abla f \cdot d\vec{r}=f(1,1,-2)-f(3,4,5)$

$=\cos(2\pi (1))+\sin(3\pi (1))+(1)(1)(-2)-(\cos(2\pi (3))+\sin(4\pi (4))+(3)(4)(5))$

$=\cos(2\pi)+\sin(3\pi)-2-(\cos(6\pi)+\sin(12\pi)+60)$

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Question

Evaluate $\int_{C} abla f \cdot d\vec{r}$ , where $f(x,y,z)=\cos(2\pi x)+\sin(3\pi y)+xyz$ , and is any path that starts at , and ends at .

Answer

Since there isn't a specific path we need to take, we just evaluate at the end points.

$\int_{C} abla f \cdot d\vec{r}=f(1,1,-2)-f(3,4,5)$

$=\cos(2\pi (1))+\sin(3\pi (1))+(1)(1)(-2)-(\cos(2\pi (3))+\sin(4\pi (4))+(3)(4)(5))$

$=\cos(2\pi)+\sin(3\pi)-2-(\cos(6\pi)+\sin(12\pi)+60)$

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Question

Evaluate $\int_{C} abla f \cdot d\vec{r}$ , where $f(x,y,z)=\cos(2\pi x)+\sin(3\pi y)+xyz$ , and is any path that starts at , and ends at .

Answer

Since there isn't a specific path we need to take, we just evaluate at the end points.

$\int_{C} abla f \cdot d\vec{r}=f(1,1,-2)-f(3,4,5)$

$=\cos(2\pi (1))+\sin(3\pi (1))+(1)(1)(-2)-(\cos(2\pi (3))+\sin(4\pi (4))+(3)(4)(5))$

$=\cos(2\pi)+\sin(3\pi)-2-(\cos(6\pi)+\sin(12\pi)+60)$

Compare your answer with the correct one above

Question

Evaluate $\int_{C} abla f \cdot d\vec{r}$ , where $f(x,y,z)=\cos(2\pi x)+\sin(3\pi y)+xyz$ , and is any path that starts at , and ends at .

Answer

Since there isn't a specific path we need to take, we just evaluate at the end points.

$\int_{C} abla f \cdot d\vec{r}=f(1,1,-2)-f(3,4,5)$

$=\cos(2\pi (1))+\sin(3\pi (1))+(1)(1)(-2)-(\cos(2\pi (3))+\sin(4\pi (4))+(3)(4)(5))$

$=\cos(2\pi)+\sin(3\pi)-2-(\cos(6\pi)+\sin(12\pi)+60)$

Compare your answer with the correct one above

Question

Evaluate $\int_{C} abla f \cdot d\vec{r}$ , where $f(x,y,z)=\cos(2\pi x)+\sin(3\pi y)+xyz$ , and is any path that starts at , and ends at .

Answer

Since there isn't a specific path we need to take, we just evaluate at the end points.

$\int_{C} abla f \cdot d\vec{r}=f(1,1,-2)-f(3,4,5)$

$=\cos(2\pi (1))+\sin(3\pi (1))+(1)(1)(-2)-(\cos(2\pi (3))+\sin(4\pi (4))+(3)(4)(5))$

$=\cos(2\pi)+\sin(3\pi)-2-(\cos(6\pi)+\sin(12\pi)+60)$

Compare your answer with the correct one above

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