Problem 49 - Calculus Single Variable

Question

The given integral \(\int_{0}^{b} f(x) d x\) represents the area of the region in the \(x y\) -plane that lies below the graph of \(f\) and above the interval \([0, b]\) of the \(x\) -axis. Express the area as an integral of the form \(\int_{c}^{d} g(y) d y .\) For example, the integral \(\int_{0}^{1} 2 x d x\) represents the area of the triangle with vertices \((0,0),(1,0),\) and \((1,2) .\) This area can also be represented as \(\int_{0}^{2}(1-y / 2) d y\). $$ \int_{0}^{4} \sqrt{x} d x $$

Step-by-Step Solution

Answer

The area can be expressed as \( \int_{0}^{2} (4 - y^2) \, dy \).

Step 1: Understand the problem

We need to express the given integral \( \int_{0}^{4} \sqrt{x} \, dx \) as a function of \( y \) by rotating or transforming the region under \( f(x) = \sqrt{x} \) over the interval \([0, 4]\) into a region described by a function of \( y \).

Step 2: Recognize the graph of \(f(x) = \sqrt{x}\)

The function \(f(x) = \sqrt{x}\) is a curve that starts at \((0,0)\) and rises to the right. The area under the curve is bounded by \(x = 0\), \(x = 4\), the \(x\)-axis, and \(y = \sqrt{x}\).

Step 3: Consider the transformation needed

We aim to express the same area in the \(xy\)-plane using an integral with respect to \(y\). This involves interpreting \(x\) in terms of \(y\). Since \(y = \sqrt{x}\), solving for \(x\) gives \(x = y^2\).

Step 4: Set bounds for \(y\)

The initial limits of integration are \(x = 0\) to \(x = 4\). These correspond to \(y = \sqrt{0} = 0\) and \(y = \sqrt{4} = 2\). Thus, the new integration limits are from \(y = 0\) to \(y = 2\).

Step 5: Create the new integral

Express the area as an integral with respect to \(y\): since \(x = y^2\), the length of the horizontal strip is \((4 - y^2)\). Thus, the integral becomes \( \int_{0}^{2} (4 - y^2) \, dy \).

Key Concepts

Definite Integral

The concept of a definite integral is crucial in understanding how to calculate the area under a curve. A definite integral is expressed as \begin{math} \int_{a}^{b} f(x) \ dx \end{math} and represents the accumulation of values of a function from a specified lower limit \( a \) to an upper limit \( b \). It essentially gives us a numerical value for the area under the curve \( f(x) \) from \( x = a \) to \( x = b \).
In practical terms, the definite integral allows us to sum continuous data over an interval. Consider it as a way of measuring the total amount of something whose rate of change you know. For instance, knowing the speed of a car over time can help you calculate the total distance traveled.

• The limits of integration, \( a \) and \( b \), define the start and endpoint of the interval over which you're summing the values.
• The function \( f(x) \) represents the quantity you are integrating.
• \( dx \) indicates that the variable of integration is \( x \).

Area Under a Curve

Understanding the concept of the area under a curve is fundamental when connecting to definite integrals. This area can represent a physical quantity like distance, probability, or any accumulated value. The integral \( \int_{0}^{b} f(x) \ dx \) not only computes the net 'signed area' in the \( xy \)-plane but also the physical area if \( f(x) \) is non-negative.
In our exercise, the integral \( \int_{0}^{4} \sqrt{x} \, dx \) represents the region under the curve of \( f(x) = \sqrt{x} \) from \( x=0 \) to \( x=4 \), and above the \( x \)-axis. This area is a geometric feature that can easily be visualized and appreciated in a graph.
The steps to find the area under a curve include:

• Setting up the integral limits \( a \) and \( b \), which denote where your curve begins and ends along the \( x \)-axis.
• Computing the integral to find the numerical value of this area. In some cases, symmetry, geometric shapes, or transformations can simplify your computations.

Remember that if \( f(x) \) goes below the \( x \)-axis, you're calculating the signed area, where areas below the axis are considered negative.

Transformation of Variables

The transformation of variables is a smart technique to simplify definite integrals by expressing them in terms of another variable. This can often make calculations more straightforward or express the original problem differently in terms of another dimension.
In the context of our exercise, we transformed an integral from \( x \) to \( y \) by using the relationship \( y = \sqrt{x} \). Solving for \( x \) gives us \( x = y^2 \), which is useful for re-expressing the area under \( f(x) \) with respect to \( y \).

• This process involves changing both the function \( f(x) \) and the bounds \( [a, b] \) to new functions and bounds in terms of \( y \).
• Such a transformation highlights new ways of viewing the problem, sometimes aligning it with other geometric interpretations like bounding ellipses or circles.

By examining transformations, you're not merely finding the area from a to b but are invited to see how regions relate through different perspectives, helping to grasp complex geometries.