Question

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer. \(x=(3-y)(y+1), x=0\)

Short answer

The area of the region is calculated by evaluating the integral from Step 5.

Step-by-step solution

Understand the Equations

We need to analyze the given equations: the first one is a parabola, rewritten as \(x = (3-y)(y+1)\), or \(x = 3y + 3 - y^2\). The second equation, \(x=0\), is the y-axis. These equations will form the boundaries of our region.

Sketch the Graphs

First, sketch the parabola from \(x = 3y + 3 - y^2\). It's a downward-opening parabola due to the \(-y^2\) term. Intersections with \(x = 0\) help us find where the parabola crosses the y-axis. Solving \(3y + 3 - y^2 = 0\) gives the intersection points, which are the roots of \(y^2 - 3y - 3 = 0\). Use the quadratic formula to find \(y\) values.

Determine Intersection Points

To find where the parabola intersects the y-axis (\(x = 0\)), solve the quadratic equation \(y^2 - 3y - 3 = 0\). Using the quadratic formula, \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a=1, b=-3, c=-3\), we calculate the roots: \(y = \frac{3 \pm \sqrt{21}}{2}\). These are the points where the parabola intersects \(x = 0\).

Set Up an Integral

The area between the parabola and the line \(x=0\) along the y-axis from \(y_1 = \frac{3 - \sqrt{21}}{2}\) to \(y_2 = \frac{3 + \sqrt{21}}{2}\) can be calculated using integration. The integral is \( \int_{y_1}^{y_2} [3y + 3 - y^2] \, dy \).

Calculate the Integral

Integrate \( 3y + 3 - y^2 \) with respect to \( y \):\[ \int (3y + 3 - y^2) \, dy = \left[ \frac{3}{2}y^2 + 3y - \frac{y^3}{3} \right]_{y_1}^{y_2} \]Evaluate this expression from \( y_1 \) to \( y_2 \) to find the area.

Estimate the Area

Estimate the area by considering the approximate shape of the region. Since the region is roughly bounded by a parabola, you can approximate it visually before you finalize your calculation with the exact integral.

Key concepts

Sketching Regions

Understanding the region where two curves meet is crucial in integration problems involving area calculation. Let's start with sketching the region for the given equations. A good sketch helps in visualizing the situation and planning the integration process.

The first equation given is a complex parabola \( x = (3-y)(y+1) \), which can be rewritten as \( x = 3y + 3 - y^2 \). This parabola opens downward due to the \(-y^2\) term. The other boundary given is \( x=0 \), which aligns with the y-axis.

By sketching these on a grid, observe the curves. The parabola will intersect the y-axis at specific points, effectively forming the bounds of our region. A careful sketch will highlight the area bounded by the parabola and the y-axis, which is crucial for the next steps.

Parabola Intersection

Intersections are vital, as they provide the limits for integration. Here, we're finding the intersection points between the parabola and the line at \( x=0 \) (the y-axis).

To calculate this, solve the quadratic equation \( y^2 - 3y - 3 = 0 \). Using the quadratic formula, \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1, b=-3, \text{ and } c=-3 \), we compute:

\[ y = \frac{3 \pm \sqrt{21}}{2} \]

This results in two intersection points along the y-axis, which are \( y_1 = \frac{3 - \sqrt{21}}{2} \) and \( y_2 = \frac{3 + \sqrt{21}}{2} \). These intersections will help define the region's top and bottom limits for integration.

Integral Setup

With the intersection points in place, setting up the integral becomes systematic. The task now is to find the area between the curve and the line using an integral.

We establish the definite integral along the y-axis from \( y_1 \) to \( y_2 \). The integral is:

\[ \int_{y_1}^{y_2} [3y + 3 - y^2] \, dy \]

This integral equation shows how the parabola's area between the y-axis from \( y_1 \) to \( y_2 \) will be calculated.

Make sure you understand where every part of the integral comes from; the limits \( y_1 \) and \( y_2 \) are the vertical bounds where the parabola intersects the y-axis.

Area Calculation

We've set up the integral, so the next step is calculating it to find the area between the curves. This integral gives a concrete number representing the area, essentially using calculus to find the space under a curve.

Perform the integration:
\[ \int (3y + 3 - y^2) \, dy = \left[ \frac{3}{2}y^2 + 3y - \frac{y^3}{3} \right]_{y_1}^{y_2} \]

This result must be evaluated from \( y_1 \) to \( y_2 \). Doing so provides the precise area of the region bounded between the parabola and the y-axis.

For a sanity check, visually approximate the shape to ensure the calculated area makes sense. Estimating the area by viewing the graph helps confirm that your integral calculations align with the visual representation.