The volume of the region R bounded by the x-axis is: $\mathbf{\iint_R(x^2+y^2)dA = \int ^{tan^{-1}(4)}_{0} \int^{\frac{2}{cos \theta}}_{0} \ r^3 dr d\theta}$

<h3>What is the volume of the solid (R) on the X-axis?</h3>

If the axis of revolution is the boundary of the plane region and the cross-sections are parallel to the line of revolution, we may use the polar coordinate approach to calculate the volume of the solid.

From the given graph:

The given straight line passes through two points (0,0) and (2,8). Thus, the equation of the straight line becomes:

$\mathbf{y-y_1 = \dfrac{y_2-y_1}{x_2-x_1}(x-x_1)}$

here:

(x₁, y₁) and (x₂, y₂) are two points on the straight line

Suppose we assign (x₁, y₁) = (0, 0) and (x₂, y₂) = (2, 8) from the graph, we have:

$\mathbf{y-0 = \dfrac{8-0}{2-0}(x-0)}$

y = 4x

Now, our region bounded by the three lines are:

y = 0
x = 2
y = 4x

Similarly, the change in polar coordinates is:

x = rcosθ,
y = rsinθ

where;

x² + y² = r² and dA = rdrdθ

Therefore;

rsinθ = 0 i.e. r = 0 or θ = 0
rcosθ = 2 i.e. r = 2/cosθ
rsinθ = 4(rcosθ) ⇒ tan θ = 4; θ = tan⁻¹ (4)

⇒ r = 0 to r = 2/cosθ
θ = 0 to θ = tan⁻¹ (4)

Then:

$\mathbf{\iint_R(x^2+y^2)dA = \int ^{tan^{-1}(4)}_{0} \int^{\frac{2}{cos \theta}}_{0} \ r^2 (rdr d\theta )}$

$\mathbf{\iint_R(x^2+y^2)dA = \int ^{tan^{-1}(4)}_{0} \int^{\frac{2}{cos \theta}}_{0} \ r^3 dr d\theta}$

Learn more about the determining the volume of solids bounded by region R here:

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3 0

1 year ago

(-4)= ? a. -4 B. 4 C.0 d. none above

Anit [1.1K]

If it’s just (-4) with nothing in front the answer is -4

3 0

2 years ago

If six girls can finish sweeping the floor of a hall in 45mins,how long will the same work take ten girls to do

AleksandrR [38]

Answer:

15 minutes i think

Step-by-step explanation:

45÷6=7.5

meaning 7.5 minutes per girl

so 4 girls would be 30 minutes of work

so 45-30=15

4 0

2 years ago