Question

Find the area bounded by y= −x 5, y= √ x 1 and the x-axis. submit your answer in fractional form

Answer 1

The area bounded by the curves
`\(y = -x^5\)`,
`\(y = √(x)\)`, and the x-axis is
`\((1)/(4)\)` square units.

Step-by-step explanation:

The first step in finding the area between curves is to identify the points of intersection. Setting the two functions equal to each other allows us to find these points:

`{split} -x^5 &= √(x) \\ -x^(10) &= x \\ x^(10) + x &= 0 \\ x(x^9 + 1) &= 0 \end{split}\]`

This gives us two solutions: (x = 0) and (x = -1). To find the bounds of integration, we need to determine where
`\(-x^5\)` is greater than or equal to
`\(√(x)\)`. The inequality
`\(-x^5 \geq √(x)\)` is satisfied for
`\(x \leq 0\)`. Therefore, the bounds of integration are from (x = -1) to (x = 0).

Next, we set up the definite integral to find the area:

`\[A = \int_(-1)^(0) (-x^5 - √(x)) \,dx\]`

Evaluating this integral gives the final answer:

`\[A = \left[-(1)/(6)x^6 - (2)/(3)x^(3/2)\right]_(-1)^(0) = (1)/(4)\]`

The area bounded by the given curves and the x-axis is
`\((1)/(4)\)` square units.

In summary, we found the points of intersection, determined the bounds of integration, set up the definite integral, and solved it to find that the area is
`\((1)/(4)\)` square units.

Answer 2

To find the bounded area, we need to determine the x-values where y = -x + 5 and y = √x + 1 intersect, and then integrate the difference of the functions across those x-values, presenting the area in fractional form.

Step-by-step explanation:

To find the area bounded by the curves y = -x + 5, y = √x + 1, and the x-axis, we need to solve for the points of intersection of the curves and then integrate between those points. Using the equality y = √x + 1 and y = -x + 5, we can find the x-values where the curves intersect. After finding the intersection points, we set up an integral from the lower x-value to the upper x-value of the difference between the functions (√x + 1) - (-x + 5).

Let's solve for the intersection points by setting the two expressions for y equal to each other:

• √x + 1 = -x + 5
• √x = -x + 4
• Squaring both sides to eliminate the square root: x + 1 = x² - 8x + 16
• This simplifies to the quadratic equation: x² - 9x + 15 = 0
• Using the quadratic formula or factoring, we find the points of intersection.

Finally, we integrate the difference of the functions to find the area, remembering to express the final result in fractional form as requested.