Find the area of the region described.
The region bounded by and
The area of the region is . (Type an exact answer.)

Ask by Barnett Craig. in the United States

Dec 10,2024

Question

Find the area of the region described. The region bounded by $$ y=1,458 \sqrt{x} $$ and $$ y=54 x^{2} $$ The area of the region is $$ \square $$. (Type an exact answer.)

UpStudy AI · Accepted Answer

To find the area of the region bounded by the curves $$ y=1,458 \sqrt{x} $$ and $$ y=54 x^{2} $$, we need to determine the intersection points of these two curves and then calculate the area between them.

First, let's find the intersection points by setting the two equations equal to each other:

$$ 1,458 \sqrt{x} = 54 x^{2} $$

Now, let's solve this equation to find the intersection points.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$1458\sqrt{x}=54x^{2}$$
- step1: Find the domain:
  $$1458\sqrt{x}=54x^{2},x\geq 0$$
- step2: Divide both sides:
  $$27\sqrt{x}=x^{2}$$
- step3: Rewrite the expression:
  $$\sqrt{x}=\frac{x^{2}}{27}$$
- step4: Raise both sides to the $$2$$-th power$$:$$
  $$\left(\sqrt{x}\right)^{2}=\left(\frac{x^{2}}{27}\right)^{2}$$
- step5: Evaluate the power:
  $$x=\frac{x^{4}}{729}$$
- step6: Cross multiply:
  $$x\times 729=x^{4}$$
- step7: Simplify the equation:
  $$729x=x^{4}$$
- step8: Move the expression to the left side:
  $$729x-x^{4}=0$$
- step9: Factor the expression:
  $$x\left(729-x^{3}\right)=0$$
- step10: Separate into possible cases:
  $$\begin{align}&x=0\&729-x^{3}=0\end{align}$$
- step11: Solve the equation:
  $$\begin{align}&x=0\&x=9\end{align}$$
- step12: Check if the solution is in the defined range:
  $$\begin{align}&x=0\&x=9\end{align},x\geq 0$$
- step13: Find the intersection:
  $$\begin{align}&x=0\&x=9\end{align}$$
- step14: Check the solution:
  $$\begin{align}&x=0\&x=9\end{align}$$
- step15: Rewrite:
  $$x_{1}=0,x_{2}=9$$
The intersection points are $$ x=0 $$ and $$ x=9 $$.

Now, we need to calculate the area between the curves from $$ x=0 $$ to $$ x=9 $$. The area can be calculated using the formula:

$$ \text{Area} = \int_{a}^{b} (f(x) - g(x)) \, dx $$

where $$ f(x) = 1,458 \sqrt{x} $$ and $$ g(x) = 54 x^{2} $$, and $$ a=0 $$ and $$ b=9 $$.

Let's calculate the area.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{0}^{9} 1458\sqrt{x}-54x^{2} dx$$
- step1: Simplify:
  $$\int_{0}^{9} 1458x^{\frac{1}{2}}-54x^{2} dx$$
- step2: Evaluate the integral:
  $$\int 1458x^{\frac{1}{2}}-54x^{2} dx$$
- step3: Use properties of integrals:
  $$\int 1458x^{\frac{1}{2}} dx+\int -54x^{2} dx$$
- step4: Evaluate the integral:
  $$972x^{\frac{3}{2}}+\int -54x^{2} dx$$
- step5: Evaluate the integral:
  $$972x^{\frac{3}{2}}-18x^{3}$$
- step6: Return the limits:
  $$\left(972x^{\frac{3}{2}}-18x^{3}\right)\bigg |_{0}^{9}$$
- step7: Calculate the value:
  $$13122$$
The area of the region bounded by the curves $$ y=1,458 \sqrt{x} $$ and $$ y=54 x^{2} $$ is $$ 13122 $$ square units.
The area of the region is 13122 square units.

Find the area of the region described.
The region bounded by and
The area of the region is . (Type an exact answer.)

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