Question

Areas of regions Find the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the given interval. $$f(x)=x^{2}-25 \text { on }[2,4]$$

Short answer

Answer: The area of the region is \(\frac{94}{3}\) square units.

Step-by-step solution

Determine the integral of the function

First, we find the integral of the given function \(f(x)=x^2 - 25\). The integral of \(x^2\) is \(\frac{1}{3}x^3\) and the integral of \(-25\) is \(-25x\). So, the integral of the given function is: $$ \int f(x) \, dx = \int (x^2 - 25) \, dx = \frac{1}{3}x^3 - 25x + C $$

Calculate the definite integral on the given interval

Now, we calculate the definite integral of the function on the interval \([2, 4]\): $$ \int_{2}^{4} (x^2 - 25) \, dx = \left[\frac{1}{3}x^3 - 25x\right]_{2}^{4} $$ To find the definite integral, we evaluate the integral at the upper bound \(x=4\) and subtract the evaluation at the lower bound \(x=2\).

Evaluate the integral at the bounds

Evaluate the integral at the upper bound \(x=4\): $$ \frac{1}{3}(4)^3 - 25(4) = \frac{1}{3}(64) - 100 = \frac{64}{3} - 100 $$ Evaluate the integral at the lower bound \(x=2\): $$ \frac{1}{3}(2)^3 - 25(2) = \frac{1}{3}(8) - 50 = \frac{8}{3} - 50 $$

Calculate the definite integral

Now, we subtract the lower bound evaluation from the upper bound evaluation: $$ (\frac{64}{3} - 100) - (\frac{8}{3} - 50) = \frac{64 - 8}{3} - (100 - 50) = \frac{56}{3} - 50 $$

Find the absolute value of the definite integral

Since the function is negative on the interval \([2, 4]\), we take the absolute value of the definite integral to find the area of the bounded region: $$ \text{Area} = \left|\frac{56}{3} - 50\right| = \left|-\frac{94}{3}\right| = \frac{94}{3} $$ So, the area of the region bounded by the graph of \(f(x) = x^2 - 25\) and the \(x\)-axis on the interval \([2, 4]\) is \(\frac{94}{3}\) square units.

Key concepts

Area under a Curve

The area under a curve is an essential concept in calculus, especially when calculating the space between a function's graph and the x-axis over a specific interval. Imagine a curve on a graph. The area under this curve is the region between the curve itself and the x-axis within certain bounds. This area can help us understand various physical quantities, such as distance or mass.

To calculate this area, integrals come into play. By performing integration, we transform a function into a shape whose area we can easily calculate. The "area under the curve" isn't always a literal area you can see. Instead, it's a mathematical concept used to find values with real-world significance, like accumulated quantities over time.

Integral Calculus

Integral calculus is a branch of calculus focused mainly on the accumulation of quantities, like areas under curves and other concepts. It offers a toolset for determining the total amount of something, whether it's space, mass, or any other cumulative measurement.

By taking the integral of a function, we can find the total values over an interval. In our exercise, we took the integral of the function \(f(x) = x^2 - 25\). The integral has both an indefinite form and a definite form.

• Indefinite Integrals: This type of integral, including a constant \(C\), represents an antiderivative of the function. It gives a family of curves.
• Definite Integrals: These compute a specific numerical value that represents the net area under the curve of a function from one point to another.

When finding areas, we use definite integrals, which help us determine the net space between the graph and the x-axis over specific intervals.

Bounded Regions

Bounded regions refer to the specific parts of the area that are enclosed within particular limits on a graph, such as x-axis boundaries. In many practical scenarios, we're interested in finding the area of these confined regions, because they often represent tangible quantities, such as material or energy within boundaries.

In the exercise provided, the bounded region is determined by the curve of \(f(x) = x^2 - 25\) and the x-axis within the interval \([2, 4]\). This region essentially represents what we need to calculate using integrals. To determine the area of this bounded region, the definite integral is taken over the set interval. This process helps in understanding how much "stuff" (like area, mass, or resources) is contained within these bounds. By performing these calculations, engineers, scientists, and mathematicians can make precise predictions and analyses in their respective fields.