Question

Compute the double integral ∬D(74xy−x²)dA over the region bounded below by y=x², above by y= √x.(Use symbolic notation and fractions where needed.) ∬D(74xy−x²)dydx=

Answer 1

The double integral ∬D(74xy−x²)dA over the region bounded below by y=x², above by y= √x is:

`\[ \int_0^1 \int_(x^2)^(√(x)) (74xy - x^2) \,dy\,dx = (19)/(21) \]`

Step-by-step explanation:

The given double integral is over the region D bounded below by
`\(y=x^2\) and above by \(y=√(x)\).` To evaluate this double integral, we integrate with respect to y first and then with respect to x. The inner integral is with respect to y, ranging from
`\(x^2\) to \(√(x)\), and the integrand is \(74xy - x^2\).`The result of the inner integral is then integrated with respect to x from 0 to 1.

To solve the inner integral, we evaluate
`\( \int_(x^2)^(√(x)) (74xy - x^2) \,dy \)`with respect to y. This involves finding the antiderivative of
`\(74xy - x^2\)`with respect to y and then evaluating it at the upper and lower bounds. After integrating, we substitute
`\(√(x)\) and \(x^2\)`into the antiderivative and subtract the result. The resulting expression in terms of x is then integrated from 0 to 1.

After performing the integrations, we arrive at the final answer
`\((19)/(21)\).` This represents the numerical value of the double integral over the specified region. It's crucial to follow the order of integration and carefully evaluate each step to obtain an accurate result.

Answer 2

`\(\iint_D (74xy - x^2) \, dy \, dx = (1277)/(210)\).`

To compute the double integral
`\(\iint_D (74xy - x^2) \, dA\)`over the region bounded below by
`\(y = x^2\) and above by \(y = √(x)\),`we follow these steps:

Step 1: Understand the Region of Integration

The region D is bounded by the curves
`\(y = x^2\) and \(y = √(x)\).`These curves intersect where
`\(x^2 = √(x)\),` which simplifies to
`\(x^4 = x\) or \(x^3 = 1\).`Therefore, the points of intersection are x = 0 and x = 1.

Step 2: Set Up the Integral

The double integral is set up as:

`\[ \iint_D (74xy - x^2) \, dA \]`

Since D is bounded by
`\(y = x^2\) below and \(y = √(x)\)`above, and x ranges from 0 to 1, the integral in terms of
`\(dydx\)`becomes:

`\[ \int_(0)^(1) \left( \int_(x^2)^(√(x)) (74xy - x^2) \, dy \right) dx \]`

Step 3: Compute the Inner Integral

First, we integrate with respect to y while treating x as a constant:

`\[ \int_(x^2)^(√(x)) (74xy - x^2) \, dy \]`

Step 4: Compute the Outer Integral

Then, we integrate the result of the inner integral with respect to xfrom 0 to 1.

Let's perform these calculations.

The value of the double integral
`\(\iint_D (74xy - x^2) \, dA\) over the specified region is \((1277)/(210)\).`