Find the area bounded by the graphs of the indicated equations over the given
interval (when stated). Compute answers to three decimal places.

The area, calculated to three decimal places, is square units.

Ask by Hill Watkins. in the United States

Nov 17,2024

Question

Find the area bounded by the graphs of the indicated equations over the given
interval (when stated). Compute answers to three decimal places.

The area, calculated to three decimal places, is square units.

Ask by Hill Watkins. in the United States

Nov 17,2024

UpStudy AI · Accepted Answer

To find the area bounded by the graphs of the equations $$y = 3x^2$$ and $$y = 27$$ over the given interval, we need to determine the points of intersection of the two graphs and then calculate the area between them.

1. Find the points of intersection:
Set the two equations equal to each other:
$$ 3x^2 = 27 $$

Solve for $$x$$:
$$ x^2 = 9 $$
$$ x = \pm 3 $$

So, the points of intersection are at $$x = -3$$ and $$x = 3$$.

2. Calculate the area between the graphs:
The area between the graphs is given by the integral of the difference between the two functions over the interval $$[-3, 3]$$.

$$ \text{Area} = \int_{-3}^{3} (27 - 3x^2) \, dx $$

Now, we will calculate the integral to find the area.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{-3}^{3} 27-3x^{2} dx$$
- step1: Evaluate the integral:
  $$\int 27-3x^{2} dx$$
- step2: Use properties of integrals:
  $$\int 27 dx-\int 3x^{2} dx$$
- step3: Evaluate the integral:
  $$27x-x^{3}$$
- step4: Return the limits:
  $$\left(27x-x^{3}\right)\bigg |_{-3}^{3}$$
- step5: Calculate the value:
  $$108$$
The area bounded by the graphs of the equations $$y = 3x^2$$ and $$y = 27$$ over the interval $$[-3, 3]$$ is 108 square units.
The area bounded by the graphs is 108 square units.

Find the area bounded by the graphs of the indicated equations over the given
interval (when stated). Compute answers to three decimal places.

The area, calculated to three decimal places, is square units.

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Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places. The area, calculated to three decimal places, is square units.

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Solution

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Find the area bounded by the graphs of the indicated equations over the given
interval (when stated). Compute answers to three decimal places.

The area, calculated to three decimal places, is square units.