Question

Constants in integrals Use the definition of the definite integral to justify the property \(\int_{a}^{b} c f(x) d x=c \int_{a}^{b} f(x) d x,\) where \(f\) is continuous and \(c\) is a real number.

Short answer

Short Answer Question: Justify the property of constants in integrals, which is given by \(\int_{a}^{b} c f(x) dx = c \int_{a}^{b} f(x) dx\), where \(f\) is a continuous function and \(c\) is a real number.

Step-by-step solution

Define the definite integral using Riemann sums.

Recall the definition of the definite integral using Riemann sums: \(\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x_i\), where \(\Delta x_i = \frac{b - a}{n}\) and \(x_i^*\) is some point in the interval \([x_{i-1}, x_i]\).

Identify the given integral with the constant term.

The given integral is \(\int_{a}^{b} cf(x)dx\). Using the definition of the definite integral, we can write this integral as: \(\int_{a}^{b} cf(x)dx = \lim_{n \to \infty} \sum_{i=1}^n cf(x_i^*) \Delta x_i\).

Factor the constant out of the sum in the Riemann sum.

We know that \(c\) is a constant, so we can factor it out of the Riemann sum: \(\int_{a}^{b} cf(x)dx = \lim_{n \to \infty} c\sum_{i=1}^n f(x_i^*) \Delta x_i\).

Recognize the Riemann sum as a definite integral.

The remaining Riemann sum in the expression is just the definite integral of \(f(x)\) over the interval \([a, b]\): \(\int_{a}^{b} cf(x)dx = c \int_{a}^{b} f(x)dx\). Now, we have justified the property of constants in integrals based on the definition of the definite integral using Riemann sums.

Key concepts

Definite Integral

The definite integral of a continuous function represents the accumulated quantity, such as area or total distance, between two points on a graph. It is denoted by \( \int_a^b f(x)dx \) and is calculated from the lower limit \( a \) to the upper limit \( b \). Essentially, it measures the net area under the curve of \( f(x) \) from \( a \) to \( b \) – that is, the area above the x-axis minus the area below the x-axis.

The process involves slicing the area into vertical strips, finding the area of each strip, and then adding them up as the number of strips approaches infinity; this is where Riemann sums come into play. When a function is multiplied by a constant, \( c \) as in our exercise, the result of the integral is simply that constant times the integral of the function without the constant. This property is intuitive if you consider that multiplying a function by a constant \( c \) just stretches or shrinks its graph vertically by that same factor.

Riemann Sums

Riemann sums are a method of approximating the definite integral of a function. They work by partitioning the interval of integration into smaller subintervals, then summing up the areas of rectangles with heights determined by the function values at chosen points within these subintervals.

In mathematical notation, a Riemann sum for our function \( f(x) \) can be written as \( \sum_{i=1}^n f(x_i^*) \Delta x_i \) where \( \Delta x_i \) is the width of each subinterval and \( x_i^* \) is the sample point in the \( i \)th subinterval. By taking the limit as \( n \) approaches infinity, we get the exact area, transitioning from a Riemann sum to a definite integral.

Properties of Integrals

Integrals have several properties that simplify their computation. One of these properties involves constants. If a function \( f(x) \) is multiplied by a constant \( c \) and integrated over an interval \( [a, b] \), the constant can be factored out of the integral; mathematically, \( \int_a^b c f(x)dx = c \int_a^b f(x)dx \).

This is known as the constant multiple rule and is what our exercise demonstrates. The property holds since each term in the Riemann sum approximation to the integral is multiplied by the same constant \( c \)– thus, the constant can be pulled out in front of the summation. Similarly, when dealing with the sum or difference of two functions being integrated, each function can be integrated separately and then added or subtracted, which is known as the sum (or difference) rule.

Limit of a Sequence

The limit of a sequence is the value that the sequence of terms approaches as the number of terms increases indefinitely. In our integral context, the limit is what transforms a Riemann sum into a definite integral. As we let the number of subdivisions \( n \) approach infinity (making the width of each rectangle infinitesimally small), the sum of the areas of the rectangles (Riemann sum) becomes an exact measurement of the area under the curve — the definite integral.

This process is encapsulated in the expression \( \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x_i \), where the limit signifies the movement towards an infinite number of increasingly narrower rectangles. Understanding the concept of limits is crucial as many calculus concepts, including the definition of derivatives and the calculation of definite integrals, rely on limits.