QUESTION 4 [TOTAL MARKS: 5 MARKS]
A manufacturing company wants to calculate the volume of a cylindrical fuel tank with a parabolic cross-section defined by and bounded between and . Using integration, evaluate the volume of the tank.
[5 marks]

Ask by Martinez Watkins. in Brunei

Mar 22,2025

Question

QUESTION 4 [TOTAL MARKS: 5 MARKS]
A manufacturing company wants to calculate the volume of a cylindrical fuel tank with a parabolic cross-section defined by and bounded between and . Using integration, evaluate the volume of the tank.
[5 marks]

Ask by Martinez Watkins. in Brunei

Mar 22,2025

UpStudy AI · Accepted Answer

To find the volume of the cylindrical fuel tank with a parabolic cross-section defined by

between

and

, we can use the formula for the volume of revolution. The volume

can be calculated using the integral:

Substituting

:

Now, calculate the integral:

Thus, the volume of the cylindrical tank is:

$$ V = \pi \int_{a}^{b} [f(x)]^2 dx $$

where:
- $$ V $$ is the volume of the solid,
- $$ f(x) $$ is the function that defines the cross-section of the solid,
- $$ a $$ and $$ b $$ are the limits of integration.

In this case, the function that defines the cross-section is $$ y=x^{2} $$, and the limits of integration are $$ x=-1 $$ and $$ x=1 $$.

Substitute the function and limits into the formula:

$$ V = \pi \int_{-1}^{1} (x^{2})^2 dx $$

Now, we can evaluate the integral to find the volume of the tank.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\pi \times \int_{-1}^{1} \left(x^{2}\right)^{2} dx$$
- step1: Multiply the exponents:
  $$\pi \times \int_{-1}^{1} x^{2\times 2} dx$$
- step2: Multiply the numbers:
  $$\pi \times \int_{-1}^{1} x^{4} dx$$
- step3: Evaluate the integral:
  $$\pi \times \frac{2}{5}$$
- step4: Multiply:
  $$\frac{\pi \times 2}{5}$$
- step5: Multiply:
  $$\frac{2\pi }{5}$$
The volume of the cylindrical fuel tank with a parabolic cross-section defined by $$ y=x^{2} $$ and bounded between $$ x=-1 $$ and $$ x=1 $$ is $$ \frac{2\pi}{5} $$ cubic units.
The volume of the cylindrical fuel tank is $$ \frac{2\pi}{5} $$ cubic units.

QUESTION 4 [TOTAL MARKS: 5 MARKS]
A manufacturing company wants to calculate the volume of a cylindrical fuel tank with a parabolic cross-section defined by and bounded between and . Using integration, evaluate the volume of the tank.
[5 marks]

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QUESTION 4 [TOTAL MARKS: 5 MARKS] A manufacturing company wants to calculate the volume of a cylindrical fuel tank with a parabolic cross-section defined by and bounded between and . Using integration, evaluate the volume of the tank. [5 marks]

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QUESTION 4 [TOTAL MARKS: 5 MARKS]
A manufacturing company wants to calculate the volume of a cylindrical fuel tank with a parabolic cross-section defined by and bounded between and . Using integration, evaluate the volume of the tank.
[5 marks]