Let R be the region in the first quadrant of the xy-plane bounded by the hyperbolas xyequals1, xyequals9, and the lines yequalsx, yequals4x. Use the transformation x equals StartFraction u Over v EndFraction , y equals uv with ugreater than0 and vgreater than0 to rewrite the integral below over an appropriate region G in the uv-plane. Then evaluate the uv-integral over G.

The area can be written as

And the value of it is approximately 1.8117

Step-by-step explanation:

x = u/v

y = uv

Lets analyze the lines bordering R replacing x and y by their respective expressions with u and v.

x*y = u/v * uv = u², therefore, x*y = 1 when u² = 1. Also x*y = 9 if and only if u² = 9x=y only if u/v = uv, And that only holds if u = 0 or 1/v = v, and 1/v = v if and only if v² = 1. Similarly y = 4x if and only if 4u/v = uv if and only if v² = 4

Therefore, u² should range between 1 and 9 and v² ranges between 1 and 4. This means that u is between 1 and 3 and v is between 1 and 2 (we are not taking negative values).

Lets compute the partial derivates of x and y over u and v

Therefore, the Jacobian matrix is

and its determinant is u/v - uv * ln(v) = u * (1/v - v ln(v))

In order to compute the integral, we can find primitives for u and (1/v-v ln(v)) (which can be separated in 1/v and -vln(v) ). For u it is u²/2. For 1/v it is ln(v), and for -vln(v) , we can solve it by using integration by parts:

Therefore,

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Step-by-step explanation:

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