Chapter 1: Problem 28
Write an integral to express the area under the graph of \(y=e^{t}\) between \(t=0\) and \(t=\ln x\), and evaluate the integral.
Short Answer
Expert verified
The area is expressed as \(x - 1\).
Step by step solution
01
Understand the Problem
We need to express the area under the curve of the function \( y = e^t \) from \( t=0 \) to \( t= \ln x \). This involves setting up an integral with the given limits of integration.
02
Set Up the Integral
The integral to express the area under the curve \( y = e^t \) from \( t = 0 \) to \( t = \ln x \) is written as:\[ \int_{0}^{\ln x} e^t \, dt \]
03
Evaluate the Integral
To evaluate the integral \( \int e^t \, dt \), recognize that the antiderivative of \( e^t \) is also \( e^t \). Then apply the limits of integration:\[ \left[ e^t \right]_{0}^{\ln x} = e^{\ln x} - e^0 \]
04
Simplify the Result
Use the property of exponents \( e^{\ln x} = x \) and \( e^0 = 1 \) to simplify the expression:\[ x - 1 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
Definite integrals are a fundamental concept in calculus used to find the area under a curve over a specific interval. Unlike indefinite integrals, which represent a broad family of antiderivatives, definite integrals yield a specific numerical value. This value corresponds to the net area enclosed by the curve and the x-axis across the given limits.
When you look at a definite integral, such as \( \int_{a}^{b} f(x) \, dx \), the limits \( a \) and \( b \) are crucial. These limits define where the measurement starts and ends. The function \( f(x) \) is the curve you are evaluating.
Evaluation involves finding the antiderivative of the function, then applying both limits using the fundamental theorem of calculus. You subtract the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit. This process will give you the area between the curve and the axis over the defined interval.
When you look at a definite integral, such as \( \int_{a}^{b} f(x) \, dx \), the limits \( a \) and \( b \) are crucial. These limits define where the measurement starts and ends. The function \( f(x) \) is the curve you are evaluating.
Evaluation involves finding the antiderivative of the function, then applying both limits using the fundamental theorem of calculus. You subtract the value of the antiderivative at the lower limit from the value of the antiderivative at the upper limit. This process will give you the area between the curve and the axis over the defined interval.
Exponential Functions
Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. In its most common form, an exponential function looks like \( y = a^x \), where \( a \) is a constant, and \( x \) is the variable.
However, when dealing with calculus, the base \( e \) becomes especially important. The natural exponential function, expressed as \( y = e^x \), is widely used because its rate of growth or decay is proportional to its current value. This characteristic makes it vital in many scientific calculations, including those linked to population growth and radioactive decay.
In our exercise, the expression \( y = e^t \) indicates exponential growth. When integrating, the antiderivative of \( e^t \) turns out to be the function itself, simplifying calculations. This unique property makes exponential functions easy to work with in calculus.
However, when dealing with calculus, the base \( e \) becomes especially important. The natural exponential function, expressed as \( y = e^x \), is widely used because its rate of growth or decay is proportional to its current value. This characteristic makes it vital in many scientific calculations, including those linked to population growth and radioactive decay.
In our exercise, the expression \( y = e^t \) indicates exponential growth. When integrating, the antiderivative of \( e^t \) turns out to be the function itself, simplifying calculations. This unique property makes exponential functions easy to work with in calculus.
Antiderivative
An antiderivative, often called the indefinite integral, is a function that represents the original function from which a derivative was taken. In the inverse operation of differentiation, finding an antiderivative involves "undoing" the derivative.
Simply put, if you have a function \( f(x) \) and you find its derivative, then the antiderivative of \( f'(x) \) will return you back to \( f(x) \), plus a constant.
In our problem, \( e^t \) is looking for its antiderivative. The beauty of the exponential function is that its antiderivative is itself \( e^t \). Hence, integrating \( e^t \) doesn't change its expression; rather it establishes a framework to evaluate definite integrals over specified limits. Calculating definite integrals utilizes this antiderivative to find the actual area represented by the integral.
Simply put, if you have a function \( f(x) \) and you find its derivative, then the antiderivative of \( f'(x) \) will return you back to \( f(x) \), plus a constant.
In our problem, \( e^t \) is looking for its antiderivative. The beauty of the exponential function is that its antiderivative is itself \( e^t \). Hence, integrating \( e^t \) doesn't change its expression; rather it establishes a framework to evaluate definite integrals over specified limits. Calculating definite integrals utilizes this antiderivative to find the actual area represented by the integral.
Limits of Integration
Limits of integration define where the calculation for the area under the curve begins and ends in definite integrals. You set these limits as the lower and upper bounds of the integral.
These bounds are vital because they provide context to the problem. Without them, you cannot evaluate a definite integral. In the problem provided, \( t = 0 \) and \( t = \ln x \) define the section of the curve we are interested in.
When evaluating the definite integral \( \int_{0}^{\ln x} e^t \, dt \), you insert these limits into the antiderivative. By substituting the upper limit and then the lower limit into the antiderivative and taking the difference, you find the net area between these bounds. Thus, the limits of integration guide and complete your evaluation of the definite integral.
These bounds are vital because they provide context to the problem. Without them, you cannot evaluate a definite integral. In the problem provided, \( t = 0 \) and \( t = \ln x \) define the section of the curve we are interested in.
When evaluating the definite integral \( \int_{0}^{\ln x} e^t \, dt \), you insert these limits into the antiderivative. By substituting the upper limit and then the lower limit into the antiderivative and taking the difference, you find the net area between these bounds. Thus, the limits of integration guide and complete your evaluation of the definite integral.
