Question

Find the area between \(y=(x-2) e^{x}\) and the \(x\) -axis from \(x=2\) to \(x=5\). (Express the answer in exact form.)

Short answer

The area is \( 2e^{5} + e^{2} \).

Step-by-step solution

Introduce the Problem

We need to find the area between the curve given by the function \( y = (x-2) e^{x} \) and the \( x \)-axis from \( x = 2 \) to \( x = 5 \). This is done by integrating the function over the specified interval.

Set Up the Integral

The area under the curve from \( x=2 \) to \( x=5 \) is found by evaluating the definite integral: \[ A = \int_{2}^{5} (x-2) e^{x} \, dx \] This integral will give us the area between the function and the \( x \)-axis, assuming the function is above the \( x \)-axis over this interval.

Apply Integration by Parts

To solve \( \int (x-2) e^{x} \, dx \), we use integration by parts: Choose \( u = x-2 \) and \( dv = e^{x} \, dx \). Then, \( du = dx \) and \( v = e^{x} \). Apply the formula: \[ \int u \, dv = uv - \int v \, du \] So, \[ \int (x-2) e^{x} \, dx = (x-2) e^{x} - \int e^{x} \, dx \] \[ = (x-2) e^{x} - e^{x} + C \] Now evaluate between \( x=2 \) and \( x=5 \).

Evaluate the Resulting Expressions

Evaluate the antiderivative from \(x=2\) to \(x=5\): Calculate \[ \left[ (x-2) e^{x} - e^{x} \right]_{2}^{5} \] This breaks down to: \[ \left[ (5-2)e^{5} - e^{5} \right] - \left[ (2-2)e^{2} - e^{2} \right] \] which simplifies to \[ \left( 3e^{5} - e^{5} \right) - \left( -e^{2} \right) \] \[ = 2e^{5} + e^{2} \]

Compute the Final Answer

The calculated value \( 2e^{5} + e^{2} \) is the exact form of the area under the curve from \( x = 2 \) to \( x = 5 \) between the function \( y = (x-2) e^{x} \) and the \( x \)-axis.

Key concepts

Definite Integral

In calculus, a definite integral helps us find the net area under a curve over a specific interval. It is computed as a limit of Riemann sums, providing a precise measurement of the accumulated quantity. The integral is "definite" because it has specific upper and lower bounds, which, in this case, are the points on the x-axis between which the area is measured.
For the problem involving the function \( y = (x-2) e^{x} \) from \( x=2 \) to \( x=5 \), the area is represented by:

• \[ \int_{2}^{5} (x-2) e^{x} \, dx \]

After you compute this integral, you'll get a precise value that represents the space between the curve and the x-axis over that interval. This concept is crucial in many fields like physics and engineering, where understanding how quantities accumulate is necessary.

Integration by Parts

Integration by parts is a technique that comes in handy when dealing with the product of two functions. It is based on the integration counterpart of the product rule for differentiation. The formula used is:

• \[ \int u \, dv = uv - \int v \, du \]

In this exercise, we identify parts of the function \( y = (x-2) e^{x} \) to simplify the computation. Choose \( u = x-2 \) and \( dv = e^{x} \, dx \). Then, differentiate \( u \) to find \( du = dx \), and integrate \( dv \) to find \( v = e^{x} \).
Substitute these into the integration by parts formula to simplify the integral to:

• \[ (x-2) e^{x} - \int e^{x} \, dx \]
• \[ = (x-2) e^{x} - e^{x} + C \]

This strategic method transforms an otherwise complex integral into an easier one.

Area Under Curve

The concept of the area under a curve refers to the region bounded by the graph of a function, the x-axis, and the two vertical lines corresponding to the interval endpoints. Calculating this area is a common application of definite integrals, providing insights into various phenomena such as distance, probability, and work done by a force.
For the curve \( y = (x-2) e^{x} \), we find the area from \( x=2 \) to \( x=5 \) by integrating the function over that interval. In the solution provided, after setting up and solving the definite integral, the exact area is determined as:

• \( 2e^{5} + e^{2} \)

This area calculation offers a visual insight into the behavior of the function within the specified range. It's important to note that this method becomes invaluable in sectors like economics to assess growth or in the analysis of physical quantities as they change over time.