Solve the problem.
-Find an equation for the level curve of the function $f ( x , y ) = \int _ { x } ^ { y } t d t$ that passes through the point $( 6,9 )$.
A) $y ^ { 2 } - x ^ { 2 } = 117$
B) $y ^ { 2 } - x ^ { 2 } = - 45$
C) $y ^ { 2 } - x ^ { 2 } = 45$
D) $y ^ { 2 } + x ^ { 2 } = 117$

Question

Quizplus · Accepted Answer

To find the equation for the level curve passing through $(6,9)$, we need to set $f(x,y) = k$ where $k$ is some constant level value.

$$f(x,y) = \int_{x}^{y}tdt = \frac{1}{2}(y^2-x^2) = k$$

Substituting $k=45$ (since we want the level curve passing through $(6,9)$) gives us:

$$\frac{1}{2}(y^2 - x^2) = 45$$

$$y^2 - x^2 = 90$$

$$y^2 - x^2 = 45 \cdot 2$$

Therefore, the correct choice is C.

Solve the Problem f(x,y)=∫xytdtf ( x , y ) = \int _ { x } ^ { y } t d tf(x,y)=∫xy​tdt

Solve the Problem $f ( x , y ) = \int _ { x } ^ { y } t d t$