Question

Perhaps the simplest change of variables is the shift or translation given by $u=x+c,$ where $c$ is a real number. a. Prove that shifting a function does not change the net area under the curve, in the sense that $${\int }_a^{b}f(x+c)dx={\int }_{a+c}^{b+c}f(u)du$$ b. Draw a picture to illustrate this change of variables in the case where $f(x)=\sin x,a=0,b=\pi ,$ and $c=\pi /2$

Short answer

Question: Prove that the integral of the shifted function is the same as the integral of the original function with shifted limits using the substitution method of integration. Additionally, illustrate this concept with a graph of the function f(x) = sin(x), the shifted function f(x+c) = sin(x+pi/2), and the given limits a=0 and b=pi. Answer: By using the substitution technique for integration with the variable u=x+c, we proved that the integral of the shifted function is equal to the integral of the original function with shifted limits: ∫[a,b] f(x+c) dx = ∫[a+c, b+c] f(u) du. This result is illustrated using a graph of the function f(x) = sin(x) and the shifted function f(x+c) = sin(x+pi/2) in the given interval [0, pi]. By comparing the net area under the curve for both functions, we can visually verify that the net area remains the same after shifting the function.

Step-by-step solution

Part a: Proving the equality of the integrals

Let's use the substitution technique for integration to prove the given equality. Let $$u=x+c.$$ So, $$du=dx,$$ and $$x=u-c$$ Now, let's find the new integration limits when $x=a$ and $x=b$. When $x=a$: $$u=a+c$$ When $x=b$: $$u=b+c$$ Now, we substitute $x+c$ with $u$, and $dx$ with $du$. We also replace the limits $a$ and $b$ with $a+c$ and $b+c$, respectively: $${\int }_a^{b}f(x+c)dx={\int }_{a+c}^{b+c}f(u)du$$ So we proved that the net area under the curve remains the same after shifting the function by using the substitution method of integration.

Part b: Illustrating the change of variables with a graph

Let's draw the function $f(x)=\sin x$ and the shifted function $f(x+c)=\sin (x+\pi /2)$ in the same graph, and visualize the effect of the change of variables with the given limits. 1. Sketch the function $f(x)=\sin x$ in the interval $[0,\pi ]$. 2. Sketch the function $f(x+c)=\sin (x+\pi /2)$ in the same graph, with the same interval. 3. Mark the points corresponding to the limits $a=0$, $b=\pi$, $a+c=0+\pi /2$ and $b+c=\pi +\pi /2$. 4. Show the net area under the curve for both functions in the given intervals. By comparing the two areas and the two functions, we can visually verify that the net area under the curve remains the same after shifting the function, as we have proved in Part a.

Key concepts

Shift or Translation

The concept of shift or translation in mathematics refers to moving a function horizontally on a graph without altering its shape. Consider the equation of a shift given by $u=x+c$, where $c$ is a constant. This operation is common and useful in calculus, particularly in integration and finding areas under curves. When you apply a shift to a function, you're effectively moving it left or right along the x-axis.

The significance of a shift is that it doesn't affect the net area under a curve, which can be shown through integration. For example, shifting a function by $c$ units translates the function to $f(x+c)$. While it moves on the graph, its integral from $a$ to $b$ equals that from $a+c$ to $b+c$ with respect to the new variable $u$, showing unchanged net area:
${\int }_a^{b}f(x+c)dx={\int }_{a+c}^{b+c}f(u)du$.

This is very practical for problems where a function needs to be analyzed at a different position, simplifying calculations or matching specific conditions.

Substitution Method of Integration

The substitution method of integration, sometimes known as u-substitution, is a technique used to simplify the process of integration by making a change of variables. The primary goal is to transform a given integral into a simpler one.

Here's how it works: Suppose you have an integral of the form $\int f(x)dx$. By substituting $u=g(x)$ and finding the corresponding $du$, you rewrite the integral in terms of the new variable $u$. This often makes the integral easier to evaluate.

In the context of the exercise, we let $u=x+c$ and consequently $du=dx$. The original integral ${\int }_a^{b}f(x+c)dx$ was rewritten using this substitution. The variable change also implied a change in the limits of integration from $x$-values $[a,b]$ to $u$-values $[a+c,b+c]$.

This transition demonstrates how substitution aids in proving properties of integrals, like showing that the net area remains the same after shifting the function horizontally. By transforming complex integrals into more manageable ones, substitution becomes a powerful tool in calculus.

Area Under the Curve

The area under the curve is a fundamental concept in integral calculus, referring to the region between the graph of a function and the x-axis. It provides a graphical interpretation of an integral.

Calculating this area involves determining the integral of the function over a specified interval. For the given problem, the area under $f(x)=\sin x$ from $0$ to $\pi$ represents the accumulated sum of vertical slices under the curve.

When a shift is applied using $f(x+c)$, it modifies the position of the function but not the overall area, as confirmed by equivalent integrals. This is crucial for verifying various properties of integrals — such as a periodic function's invariance under specific translations.

Graphs are a useful tool for visualizing these concepts. By graphing $f(x)=\sin x$ and its shifted version $\sin (x+\pi /2)$, one can see that while the function's position changes, the area between curves and limits remains equal, offering a practical understanding of how translations affect but don't alter net area. This reflects the geometric intuition behind overcoming complex calculus problems.