Question

Find the area bounded by \(y\left(4+25 x^{2}\right)=5, x=0, y=0\), and \(x=4 .\) Use a table of integrals or a CAS.

Short answer

The area is approximately \(7.85\) square units.

Step-by-step solution

Identify the Curve Equation

Given the equation of the curve: \(y(4 + 25x^2) = 5\). We can rewrite it to express \(y\) in terms of \(x\):\[y = \frac{5}{4 + 25x^2}\].

Determine the Bounds of Integration

We need to find the area bounded by the curve and the lines \(x=0\), \(x=4\), and \(y=0\). The limits of integration for the area are from \(x = 0\) to \(x = 4\).

Setup the Integral for Area

The required area under the curve can be expressed as an integral:\[A = \int_{0}^{4} \frac{5}{4 + 25x^2} \, dx\].

Evaluate the Integral

Using a table of integrals or a CAS, evaluate the integral:\[\int \frac{1}{4 + 25x^2} \, dx = \frac{1}{5} \cdot \frac{1}{2} \tan^{-1}\left(\frac{x}{2/5}\right)\].Apply the constant 5 from the integral and adjust the antiderivative:\[= \frac{1}{10} \tan^{-1}\left(\frac{x}{2/5}\right)\].Evaluating from \(0\) to \(4\):\[A = 5 \left[ \frac{1}{10} \tan^{-1}\left(\frac{5x}{2}\right) \right]_{0}^{4}\].

Calculate the Definite Integral

Substitute the bounds into the evaluated antiderivative:\[A = 5 \left[ \frac{1}{10} \tan^{-1}(\frac{5 \times 4}{2}) - \frac{1}{10} \tan^{-1}(\frac{5 \times 0}{2}) \right]\].Simplify the expression:\[= \frac{1}{2} \left[ \tan^{-1}(10) - \tan^{-1}(0) \right]\].

Determine the Final Area

Calculate the area using the known values of \(\tan^{-1}\):\(\tan^{-1}(10)\) is a constant that can be evaluated numerically or using approximations. Hence, the area \(A\) simplifies to:\\[\frac{5}{2} \left(\tan^{-1}(10) - 0\right)\].The final numerical value is obtained using a calculator to find \(\tan^{-1}(10)\).

Key concepts

Integral Calculus

Integral calculus is about finding the total accumulation of quantities. This could mean calculating areas, volumes, or other values that add up over a curve or surface.
In the context of the given exercise, we aim to calculate the area under a curve.
This requires the evaluation of an integral over specified limits. To understand how integral calculus works: - It deals with integrals, which are essentially the reverse process of differentiation. - While differentiation finds the rate of change, integration aggregates these changes over a range. We use integrals to sum infinitely small pieces over a continuous interval. The result gives a total accumulation or area under the curve.
This is particularly useful in various fields like physics, engineering, and economics, where such cumulative measurements are necessary.

Definite Integral

A definite integral represents the precise computation of an integral over a specific interval.
In our exercise, we need to find the definite integral of a function from one boundary to another, specifically between two values of \(x\).In mathematical notation, you would see this written as \[ \int_{a}^{b} f(x) \, dx \], where:

• \(a\) and \(b\) are the limits of integration representing the start and end values.
• \(f(x)\) is the integrand, or function to be integrated.
• \(dx\) signals integration with respect to \(x\).

Evaluating a definite integral means computing the accumulated value of a function over the specified interval.
This involves substituting the upper and lower limits into the antiderivative of the function and then subtracting these results.

Area Under Curve

Calculating the area under a curve is a common application of definite integrals.
When you think about it visually, you want to determine the space occupied beneath the curve and above the x-axis, within specified limits.To find the area under a curve:

• Identify the function representing the curve, such as \(y = f(x)\).
• Determine the bounds or limits of integration that enclose the area of interest.
• Set up the definite integral of the function over these bounds.

The result of this integral gives a numerical value representing the area, which can be a crucial component in problems involving physical quantities, economic models, or probability distributions.