Find the accumulation function F. Then evaluate F at each value of the independent variable and

Find the accumulation function F. Then evaluate F at each...

Key Concepts

Accumulation Function

An accumulation function represents the value of a quantity that has accumulated over time or space, computed as a definite integral from a fixed starting point to a variable endpoint. It captures the total effect or accumulation of a rate function, allowing the interpretation of the integral as the net area under a curve up to that point.

Definite Integral

A definite integral calculates the exact accumulated value, such as area under a curve, between two specified limits. It is a fundamental concept in calculus used to evaluate the total accumulation or net change over an interval of the independent variable.

Fundamental Theorem of Calculus

This theorem bridges the concepts of differentiation and integration, stating that the process of integrating a function and then differentiating the result recovers the original function. It also provides a method to evaluate definite integrals by using antiderivatives, which is central to forming and solving accumulation functions.

Exponential Functions

Exponential functions involve expressions where the variable appears in the exponent. They are significant in calculus due to their unique rate of growth or decay properties, and their integrals often require specific techniques. In accumulation functions, an exponential integrand leads to calculations that emphasize constant proportional change.

Graphical Interpretation of Integrals

The graphical interpretation of an integral involves visualizing the definite integral as the area under a curve between specified limits. This representation aids in understanding how the accumulation function is constructed and how changes in the independent variable adjust the total accumulated area.

Transcript

00:01 In this question, we are required to evaluate the accumulation function f for the function fy is equals to integration minus 1 to y, 4, e to the power x upon 2 dx.

00:21 After that, we are required to find the value of f for y is equals to minus 1, y is equals to 0 and y is equals to 4.

00:39 And finally, we are required to draw the graph that shows the area for f minus 1, f0 and f4.

00:54 So let's see how to solve this question.

00:58 First of all, let's integrate the above function to find the value of fy.

01:06 So fy will be equals to integration minus 1 to y 4 e to the power x upon 2 d x and the integration of 4 e to the power x upon 2 will be equal to 8 a to the power x upon 2 and the limits are minus 1 to y now substitute all the values so we get f y is equal to 8 into 8 to the power 8 into a to the power y by 2 minus a to the power minus 1 upon 2 and now let's substitute y is equals to minus 1 in the above equation so we get f minus 1 is equal to 8 into a to the power minus 1 upon 2 minus a to the power minus 1 upon 2 so we get f minus 1 is equal to 0 and and now let's draw the graph for f minus 1 and the graph is shown below.

02:26 So this is the graph for f minus 1 and f minus 1 is equal to 0 is the final answer for part a.

02:33 And now let's move to part b.

02:39 Part b, substitute y is equal to 0 in the integrated value.

02:50 So we can write f0 is equal to 8 into 8.

02:57 E to the power 0 minus e to the power minus 1 upon 2.

03:02 So this will be equal to 8 into and the value of e to the power 0 will be equals to 1 minus e to the power minus 1 upon 2...

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