Question

Find the area of the region bounded by the curve \(y=\frac{7}{x^{2}}\), the \(x\) -axis, and on the left by \(x=1\).

Short answer

The area is 7 square units.

Step-by-step solution

Identify the Area to be Calculated

To find the area under the curve \(y = \frac{7}{x^2}\) from \(x = 1\) to some point, we need to integrate this function with respect to \(x\). This area is defined to be between the curve and the x-axis.

Set up the Definite Integral

Since the problem is only bounded on the left at \(x = 1\), and assumes extending indefinitely to the right of \(x = 1\), you'll need to set an arbitrary upper bound \(x = a\). Convert this to the integral: \[\int_{1}^{a} \frac{7}{x^2} \, dx\]

Integrate the Function

Integrate the function described in the previous step. The integral of \(\frac{7}{x^2}\) is computed as: \[-\frac{7}{x}\] Add the constant limits:\[\left[-\frac{7}{x}\right]_{x=1}^{x=a}\]

Evaluate the Definite Integral from 1 to a

Now evaluate the integral using upper and lower bounds:\[-\frac{7}{a} + 7\]This expression gives the area from \(x = 1\) to \(x = a\).

Find the Total Area as a Approaches Infinity

To find the total finite area, evaluate the limit as \(a\) approaches infinity:\[\lim_{a \to \infty} \left(-\frac{7}{a} + 7\right)\]Since \(-\frac{7}{a} \to 0\), the total area is:\(7\).

Key concepts

Definite Integral

A definite integral is a mathematical tool used to compute the area under a curve within specific bounds on an axis. In calculus, the notation for a definite integral \[ \int_{a}^{b} f(x) \, dx \] indicates the integration of the function \( f(x) \) from the starting point \( a \) to the endpoint \( b \). This process provides the accumulated value or total area under the curve of \( f(x) \) from \( x = a \) to \( x = b \).

When calculating definite integrals, the limits of integration are critical as they define the section of the curve being evaluated. The value obtained represents the net area, taking into account portions above and below the x-axis.

• The fundamental theorem of calculus connects derivatives with integrals and ensures that you can compute a definite integral by finding an anti-derivative or an indefinite integral of a function.
• For the given exercise, integrating \( \frac{7}{x^2} \) from 1 to \( a \) helps in calculating the area of the region between the curve and x-axis, with one finite boundary and one extending to infinity.

Area Under Curve

Finding the area under a curve is a common problem in calculus. This area represents a physical quantity related to the integral of a given function. By integrating a function, it sums up infinitely small slices of area defined by the curve and the x-axis.

Here's how it works:

• The function describes the height of each slice. Integrating this function means adding up the areas of slices from the lower limit of integration to the upper limit.
• In the context of the problem \( y = \frac{7}{x^2} \), the area under the curve from \( x = 1 \) to \( x = a \) is considered.

The formula for the area is \[\int_{1}^{a} \frac{7}{x^2} \, dx \]. After solving this integral, a finite area value or a total area is obtained as \( a \) approaches infinity. This calculation yields meaningful physical interpretations, such as force, work done, or distance.

Improper Integral

An improper integral occurs when integrating a function over an interval with bounds that include infinity or where the function becomes unbounded. This usually happens when one or both of the integration limits are infinite or the function approaches infinity within the interval.

For the given exercise, treating the right boundary of the region as \( x = a \) with \( a \to \infty \) creates an improper integral. The integral is: \[ \lim_{a \to \infty} \int_{1}^{a} \frac{7}{x^2} \, dx \].

• First, find the indefinite integral of the function to get \(-\frac{7}{x} \).
• Then, evaluate the expression from 1 to \( a \).
• Finally, calculate the limit as \( a \to \infty \).

This process reveals that the area becomes finite and equal to 7, even as the upper boundary extends to infinity, which is an interesting feature of improper integrals in calculus.