Question

In Exercises \(47-56,\) use graphs, your knowledge of area, and the fact that \(\quad\) $$\int_{0}^{1} x^{3} d x=\frac{1}{4}$$ to evaluate the integral. $$\int_{2}^{3}(x-2)^{3} d x$$

Short answer

The value of the integral \(\int_{2}^{3}(x-2)^{3} d x\) is \(\frac{1}{4}\).

Step-by-step solution

Substitution

Let us apply the substitution \(u = x - 2\). Thus the integral \(\int_{2}^{3}(x-2)^{3} d x\) becomes \(\int_{0}^{1}u^{3} du\). This substitution maps the interval [2,3] to [0,1].

Using given information

From the given information, we know that \(\int_{0}^{1}x^{3} dx =\frac{1}{4}\). We apply this to our integral after substitution. Hence, \(\int_{0}^{1}u^{3} du = \frac{1}{4}\).

Back Substitution

Substitute back \(u = x - 2\) to regain the original variable of the problem. So, \(\int_{2}^{3}(x-2)^{3} d x = \frac{1}{4}\).

Key concepts

Definite Integral

The concept of the definite integral is fundamental to calculus and represents the accumulation of quantities, such as area. It is defined as the limit of a sum of areas of rectangles under a curve as the width of the rectangles approaches zero. Mathematically, the definite integral of a function f(x) from a to b is denoted as \( \int_{a}^{b} f(x) dx \), where a and b are the limits of integration.

When calculating the area under a curve between two points on the x-axis, we are finding the definite integral. For example, the solution provided in the exercise shows the use of a definite integral to calculate the area under the curve x^3 from 0 to 1. Knowing the definite integral allows us to apply it in different scenarios with variable changes, while still preserving the geometric interpretation as the area under a curve.

U-substitution Method

The u-substitution method is a technique used in integrating complex functions by simplifying them into a form that is easier to integrate. It is analogous to a change of variables that transforms a given integral into a standard form.

Think of it as renaming a part of the original function to make it easier to work with. This can be particularly useful when dealing with composite functions or functions within functions. In the step by step solution, we see the substitution u = x - 2, simplifying the integral of (x-2)^3 to the integral of u^3, which is known and therefore can be calculated easily. Important aspects of this method include choosing an appropriate substitution, modifying the limits of integration, and substituting back to the original variable after integrating.

Area Under the Curve

The area under the curve is a visual and practical interpretation of the definite integral. In the context of calculus, it represents the total 'accumulated' value between the curve of a function and the x-axis over an interval.

Areas can be positive or negative depending on whether the curve is above or below the x-axis. In real-world terms, this could represent physical distances, quantities of material, or other cumulative magnitudes. As illustrated in the exercise, the area under x^3 between x=0 and x=1 is being evaluated, which has a practical meaning in terms of area, despite the function not representing a geometric shape traditionally associated with easy area calculations like rectangles or triangles.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a vital principle that connects differentiation and integration, establishing an inverse relationship between the two processes. It states, in its first part, that if a function f is continuous over [a, b] and F is the indefinite integral of f on [a, b], then the definite integral of f from a to b is given by \( F(b) - F(a) \).

This theorem can be used to evaluate definite integrals without calculating limits of sums. In our example, we used a known integral to find a new one after a change of variables, showcasing how the Fundamental Theorem of Calculus is applied in solving real integration problems. It demonstrates that the act of integration, which can be seen as accumulating changes, is the reverse of differentiation, which measures instantaneous rates of change.