Problem 19 - Calculus: Early Transcendentals

Question

Express the limit as a definite integral on the given interval. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n [5(x_i^*)^3 - 4x_i^*] \, \Delta x \), [2, 7]

Step-by-Step Solution

Answer

The limit is expressed as \( \int_{2}^{7} (5x^3 - 4x) \, dx \).

Step 1: Identify the function

The given sum represents a Riemann sum which approximates the integral of a function. The function inside the sum is given by \[ f(x) = 5x^3 - 4x \] This is the function we will integrate over the given interval.

Step 2: Determine the interval

The Riemann sum is defined over the interval [2, 7]. This is the interval over which the definite integral will be calculated, so we have \[ [a, b] = [2, 7] \] with \( a = 2 \) and \( b = 7 \).

Step 3: Express the limit as a definite integral

As \( n \to \infty \), the Riemann sum \[ \sum_{i=1}^{n} \left[ 5(x_i^*)^3 - 4x_i^* \right] \Delta x \]approaches the definite integral \[ \int_{a}^{b} f(x) \, dx = \int_{2}^{7} (5x^3 - 4x) \, dx \].

Key Concepts

Riemann Sum

The Riemann sum is an essential concept in calculus that helps us approximate the value of a definite integral. It is named after the German mathematician Bernhard Riemann and provides a way to calculate areas under curves. Imagine dividing the area beneath a curve into a number of small rectangles or trapezoids. By summing the areas of these shapes, we can estimate the total area under the curve.

In the exercise provided, each term of the Riemann sum takes the form \( f(x_i^*) \Delta x \), where \( f(x_i^*) \) represents the function value at some sample point \( x_i^* \) within each subinterval. The width of the subinterval is denoted by \( \Delta x \). As the number of rectangles \( n \) increases and \( \Delta x \) becomes smaller, the approximation of the area becomes more accurate.

Limit as a Definite Integral

As we increase the number of subintervals in a Riemann sum, we take the limit as \( n \to \infty \). This means each subinterval width \( \Delta x \) becomes infinitesimally small. This process transforms our approximation into an exact calculation, known as a definite integral.

In the case of our exercise, as \( n \to \infty \), the Riemann sum converts to the definite integral \( \int_{2}^{7} (5x^3 - 4x) \, dx \). Thus, the limit signifies the transition from the sum of the finite approximations to finding the exact area under the curve of the function \( f(x) = 5x^3 - 4x \) on the interval [2, 7].

Function Integration

Integration is the mathematical process of finding the integral of a function. This involves calculating the area under the curve of a given function over a specified interval. The function's integral provides crucial information, like the total accumulated value or area, and it is denoted by \( \int f(x) \, dx \).

For the exercise at hand, the function \( f(x) = 5x^3 - 4x \) is integrated over the given interval [2, 7]. This means we are calculating the total area beneath the curve of this function, from \( x = 2 \) to \( x = 7 \). This integral represents the exact value that we initially approximated using the Riemann sum.

Interval of Integration

The interval of integration is the range over which the integration is performed. It is depicted by the bounds \( [a, b] \), where \( a \) and \( b \) are the lower and upper limits, respectively. This interval determines where our calculations start and end on the x-axis.

For the given exercise, the interval of integration is [2, 7]. This means we are interested in calculating the area under the curve \( 5x^3 - 4x \) from \( x = 2 \) to \( x = 7 \). Understanding these limits is crucial as they establish the scope of the definite integral and directly impact the resulting calculated area.