Problem 69 - Calculus Single Variable

Question

In each of Exercises 69-76, calculate the volume of the solid obtained when the region \(\mathcal{R}\) is rotated about the given line \(\ell\) \(\mathcal{R}\) is the region between \(y=6-x^{2}\) and \(y=-x ; \ell\) is the line \(x=-4\)

Step-by-Step Solution

Answer

The volume is \(252\pi\).

Step 1: Find the Intersection Points

Set the equations of the curves equal to each other to find the intersection points: \[ 6 - x^2 = -x \] Rearrange the equation: \[ x^2 - x - 6 = 0 \] Factor or use the quadratic formula to solve for \(x\). The factors are \((x-3)(x+2)\), giving the intersection points \(x = 3\) and \(x = -2\).

Step 2: Determine the Outer and Inner Radius

The radii for the washer method are calculated from the line \(x=-4\), which we will rotate around. - Outer radius: \(R = |x + 4| = x + 4\) for the curve \(y = 6 - x^2\) - Inner radius: \(r = |x + 4| = x + 4\) for the curve \(y = -x\).

Step 3: Set Up the Washer Method Formula

The volume \(V\) is given by the washer method formula: \[ V = \pi \int_{a}^{b} \left( R^2 - r^2 \right) \, dx \]Where \(R = 6 - x^2 + 4) = 10 - x^2\) and \(r = x + 4 + x = 4 + 2x\).

Step 4: Calculate the Volume Integral

Substitute the expressions for \(R\) and \(r\) into the integral:\[ V = \pi \int_{-2}^{3} [(10 - x^2)^2 - (4 + 2x)^2] \, dx \] Simplify and expand the squares:\[ V = \pi \int_{-2}^{3} (100 - 20x^2 + x^4 - (16 + 16x + 4x^2)) \, dx \]\[ = \pi \int_{-2}^{3} (x^4 - 24x^2 - 16x + 84) \, dx \]

Step 5: Integrate and Calculate

Integrate the function:\[ V = \pi \left[ \frac{x^5}{5} - 8x^3 - 8x^2 + 84x \right]_{-2}^{3} \]Calculate the definite integral:- Evaluate at \(x = 3\)- Evaluate at \(x = -2\)Subtract the results and multiply by \(\pi\) for the final volume.

Key Concepts

Washer Method

The Washer Method is a powerful technique used to find the volume of a solid of revolution. It is particularly useful when the solid has a hole in it, resembling a washer or donut shape. This name comes from the geometric figure's similarity to the everyday object known as a washer in hardware.When using the Washer Method, you revolve a region around a given line or axis and use washers to fill this volume. Each washer has two radii: an outer radius and an inner radius, which effectively help us capture the hole in the middle. Since we revolve around a vertical or horizontal line, we visualize the washer as a three-dimensional object with these two concentric circles flattened against this line.The formula for the Washer Method is: \[ V = \pi \int_{a}^{b} \left( R(x)^2 - r(x)^2 \right) \, dx \]Where

• \( R(x) \) is the outer radius
• \( r(x) \) is the inner radius
• The limits \( a \) and \( b \) are the points of intersection

To apply this method, it's crucial to correctly identify these radii and limits, ensuring the holes are correctly accounted for in the volume calculation.

Intersection Points

Finding the intersection points between curves is a vital first step in many integration problems involving areas and volumes. These points tell us the bounds of the region our solid will cover when rotated.To find where two curves intersect, set their equations equal to each other and solve for the variable. In our example:\[ 6 - x^2 = -x \]Simplifies to:\[ x^2 - x - 6 = 0 \]This is a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula. Here, we factored it:\[ (x - 3)(x + 2) = 0 \]Revealing the intersection points at \( x = 3 \) and \( x = -2 \). These points become the limits, \( a \) and \( b \), for our definite integral in the Washer Method.

Definite Integral

The definite integral is essential in determining the volume of a solid. After setting up the integral's limits using the intersection points, the definite integral calculates the accumulated value of the radii differences squared over the interval.For the Washer Method, the integral is set up as:\[ V = \pi \int_{a}^{b} \left( R(x)^2 - r(x)^2 \right) \, dx \]Here, we incorporate the functions representing the outer and inner radii:- Outer radius, \( R(x) = 10 - x^2 \)- Inner radius, \( r(x) = 4 + 2x \)These represent the distances from each curve to the axis of revolution. Evaluating this integral over the limits defined by the intersection points provides the desired volume of the solid.

Outer and Inner Radius

In volume integration via the Washer Method, determining the outer and inner radius is key to setting up your integral correctly.In our problem, the region is revolved around the vertical line \( x = -4 \). This affects how we calculate each radius:

• The **Outer Radius** is measured from the furthest curve away from the line of revolution: \( y = 6 - x^2 \). So we have:\[ R(x) = |x + 4| = x + 4 \]Ultimately written as \( R(x) = 10 - x^2 \) after adjustments to account for the fact we revolve around \( x = -4 \).
• The **Inner Radius**, closer to the line, is measured from the curve \( y = -x \):\[ r(x) = |x + 4| = x + 4 \]Which transforms to \( r(x) = 4 + 2x \).

Correctly identifying these radii ensures that each washer accounts precisely for the material that forms our solid, without leaving unfilled gaps or incorrect fills. Properly transformed, they align with the axis of revolution, critical for an accurate volume calculation.