Q. 15 - Calculus

Question

Although the definite integral of a sum of functions is equal to the sum of the definite integrals of those functions, the definite integral of a product of functions is not the product of two definite integrals.

1. Use mathematical notation to write the preceding sentence in this form:

                    _____ $=$______ , but _____ $\ne$______.

      b. Choose two simple functions $f$ and $g$ so that you can calculate the definite integrals of $f$ , $g$, and $f+g$ on $\left[0,1\right]$, and show that       the sum of the first two definite integrals is equal to the third.

       c. Find two simple functions $f$ and $g$ such that ${\int }_0^{1}f\left(x\right)g\left(x\right)dx$ is not equal to the product of ${\int }_0^{1}f\left(x\right)dx$ and ${\int }_0^{1}g\left(x\right)dx$(Hint: Choose      $f$ and $g$ so that you can calculate the definite integrals involved.)

Step-by-Step Solution

Answer

Part $\left(a\right)$: ${\int }_a^b\left(f\right(x)+g(x\left)\right)dx={\int }_a^{b}f\left(x\right)dx+{\int }_a^{b}g\left(x\right)dx$ but,${\int }_a^b\left(f\right(x)·g(x\left)\right)dx\ne {\int }_a^{b}f\left(x\right)dx·{\int }_a^{b}g\left(x\right)dx$.

Part $\left(b\right)$: The sum of the first two definite integrals is equal to the third is proved.

Part $\left(c\right)$: ${\int }_0^1\left(f\right(x)·g(x\left)\right)dx$ is not equal to the product of   ${\int }_0^{1}f\left(x\right)dx$ and ${\int }_0^{1}g\left(x\right)dx$ is proved.

Part a Step 1 . Given information

The definite integral of a sum of functions is equal to the sum of the definite integrals of those functions, the definite integral of a product of functions is not the product of two definite integrals.

Part a Step 2 . The mathematical notation is,

${\int }_a^b\left(f\right(x)+g(x\left)\right)dx={\int }_a^{b}f\left(x\right)dx+{\int }_a^{b}g\left(x\right)dx$ 

But,

${\int }_a^b\left(f\right(x)·g(x\left)\right)dx\ne {\int }_a^{b}f\left(x\right)dx·{\int }_a^{b}g\left(x\right)dx$

Part b Step 1 . The objective is to show that, ∫ a b f ( x ) d x + ∫ a b g ( x ) d x = ∫ a b ( f ( x ) + g ( x ) ) d x .

Let $f\left(x\right)=2,g\left(x\right)=x$.

Now, 

$$\begin{gathered} {\int }_a^{h}f\left(x\right)dx={\int }_0^{1}2dx \\ =2\left(1\right) \\ =2 \end{gathered}$$

Again,

$$\begin{gathered} {\int }_a^{b}g\left(x\right)dx={\int }_0^{1}xdx \\ =\frac{1}{2}\left(1\right) \\ =\frac{1}{2} \end{gathered}$$

Part b Step 2 . Hence,

$$\begin{gathered} {\int }_a^b\left(f\right(x)+g(x\left)\right)dx={\int }_a^{b}f\left(x\right)dx+{\int }_a^{b}g\left(x\right)dx \\ ={\int }_0^{1}2dx+{\int }_0^{1}xdx \\ =2\left(1\right)+\frac{1}{2} \\ =2+\frac{1}{2} \\ =\frac{5}{2} \end{gathered}$$

So,

${\int }_a^{b}f\left(x\right)dx+{\int }_a^{b}g\left(x\right)dx=2+\frac{1}{2}=\frac{5}{2}={\int }_a^b\left(f\right(x)+g(x\left)\right)dx$

Therefore, it is proved.

Part c Step 1 . The objective is to show that, src="data:image/svg+xml;base64,<svg xmlns="http://www.w3.org/2000/svg" xmlns:wrs="http://www.wiris.com/xml/mathml-extension" height="48" width="287" wrs:baseline="29"><!--MathML: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>&#x222B;</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>&#xB7;</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>x</mi><mo>&#x2260;</mo><msubsup><mo>&#x222B;</mo><mn>0</mn><mn>1</mn></msubsup><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi><mo>&#xB7;</mo><msubsup><mo>&#x222B;</mo><mn>0</mn><mn>1</mn></msubsup><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></math>--><defs><style type="text/css">@font-face{font-family:'aec8956637a99787bd197eacd77acce';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'math1ff0cdbd87de9d867973e3e6fb7';src:url(data:font/truetype;charset=utf-8;base64,AAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAERjdnQgDVUNBwAAAWAAAAA6Z2x5ZoPi2VsAAAGcAAABVWhlYWQQC2qxAAAC9AAAADZoaGVhCGsXSAAAAywAAAAkaG10eE2rRkcAAANQAAAAEGxvY2EAHTwYAAADYAAAABRtYXhwBT0FPgAAA3QAAAAgbmFtZaBxlY4AAAOUAAABn3Bvc3QB9wD6AAAFNAAAACBwcmVwa1uragAABVQAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAALciKyJg//8AAAC3IisiYP///0rd192jAAEAAAAAAAAAAAAAAVQDLACAAQAAVgAqAlgCHgEOASwCLABaAYACgACgANQAgAAAAAAAAAArAFUAgACrANUBAAErAAcAAAACAFUAAAMAA6sAAwAHAAAzESERJSERIVUCq/2rAgD+AAOr/FVVAwAAAQCAAVUA6wHAAAMAGxgBsAQQsQAD9LECA/SxBQP0ALAEELEDBvQwMRMzFSOAa2sBwGsAAQBL/2sCCwNAABkAAAEmDgEDAg4BIyY0NgYjFj4BExI+ATMWDgE2AfU1NSAgIEA2QCpVCzU1NiAgIEA1QCsBVQsDFRYri/7g/sCKIBVAFUAVK4oBIAFAiyAVQBU/AAMAgABzAtUCjwADAAcACwBlGAGwDBCwBtSwBhCwBdSwDBCwAdSwARCwANSwBhCwBzywBRCwBDywARCwAjywABCwAzwAsAwQsAbUsAYQsAfUsAcQsAHUsAEQsALUsAYQsAU8sAcQsAQ8sAEQsAA8sAIQsAM8MTATITUhHQEhNQMBMwGAAlX9qwJVf/5WVQGrAcBV1VVVAU/95AIcAAAAAAEAAAABAADVeM5BXw889QADBAD/////1joTc//////WOhNzAAD/IASAA6sAAAAKAAIAAQAAAAAAAQAAA+j/agAAF3AAAP+2BIAAAQAAAAAAAAAAAAAAAAAAAAQDUgBVAWsAgAH0AEsDVgCAAAAAAAAAACgAAABbAAAAuQAAAVUAAQAAAAQAXgAFAAAAAAACAIAEAAAAAAAEAADeAAAAAAAAABUBAgAAAAAAAAABABIAAAAAAAAAAAACAA4AEgAAAAAAAAADADAAIAAAAAAAAAAEABIAUAAAAAAAAAAFABYAYgAAAAAAAAAGAAkAeAAAAAAAAAAIABwAgQABAAAAAAABABIAAAABAAAAAAACAA4AEgABAAAAAAADADAAIAABAAAAAAAEABIAUAABAAAAAAAFABYAYgABAAAAAAAGAAkAeAABAAAAAAAIABwAgQADAAEECQABABIAAAADAAEECQACAA4AEgADAAEECQADADAAIAADAAEECQAEABIAUAADAAEECQAFABYAYgADAAEECQAGAAkAeAADAAEECQAIABwAgQBNAGEAdABoACAARgBvAG4AdABSAGUAZwB1AGwAYQByAE0AYQB0AGgAcwAgAEYAbwByACAATQBvAHIAZQAgAE0AYQB0AGgAIABGAG8AbgB0AE0AYQB0AGgAIABGAG8AbgB0AFYAZQByAHMAaQBvAG4AIAAxAC4AME1hdGhfRm9udABNAGEAdABoAHMAIABGAG8AcgAgAE0AbwByAGUAAAMAAAAAAAAB9AD6AAAAAAAAAAAAAAAAAAAAAAAAAAC5BxEAAI2FGACyAAAAFRQTsQABPw==)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'round_brackets18549f92a457f2409';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}</style></defs><text font-family="math1ff0cdbd87de9d867973e3e6fb7" font-size="32" text-anchor="middle" x="8.5" y="33">&#x222B;</text><text font-family="Arial" font-size="12" text-anchor="middle" x="14.5" y="44">0</text><text font-family="Arial" font-size="12" text-anchor="middle" x="21.5" y="11">1</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="28.5" y="29">(</text><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="33.5" y="29">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="42.5" y="29">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="48.5" y="29">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="55.5" y="29">)</text><text font-family="math1ff0cdbd87de9d867973e3e6fb7" font-size="16" text-anchor="middle" x="62.5" y="29">&#xB7;</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="70.5" y="29">g</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="79.5" y="29">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="85.5" y="29">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="92.5" y="29">)</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="97.5" y="29">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="104.5" y="29">d</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="113.5" y="29">x</text><text font-family="math1ff0cdbd87de9d867973e3e6fb7" font-size="16" text-anchor="middle" x="126.5" y="29">&#x2260;</text><text font-family="math1ff0cdbd87de9d867973e3e6fb7" font-size="32" text-anchor="middle" x="143.5" y="33">&#x222B;</text><text font-family="Arial" font-size="12" text-anchor="middle" x="149.5" y="44">0</text><text font-family="Arial" font-size="12" text-anchor="middle" x="156.5" y="11">1</text><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="163.5" y="29">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="172.5" y="29">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="178.5" y="29">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="185.5" y="29">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="192.5" y="29">d</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="201.5" y="29">x</text><text font-family="math1ff0cdbd87de9d867973e3e6fb7" font-size="16" text-anchor="middle" x="210.5" y="29">&#xB7;</text><text font-family="math1ff0cdbd87de9d867973e3e6fb7" font-size="32" text-anchor="middle" x="222.5" y="33">&#x222B;</text><text font-family="Arial" font-size="12" text-anchor="middle" x="228.5" y="44">0</text><text font-family="Arial" font-size="12" text-anchor="middle" x="235.5" y="11">1</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="243.5" y="29">g</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="252.5" y="29">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="258.5" y="29">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="265.5" y="29">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="272.5" y="29">d</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="281.5" y="29">x</text></svg>" role="math" localid="1648702128437" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/fbbc2f75-e230-4075-aab0-04e2451764e0.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220331%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220331T050305Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=1152868ecc0d9f7ab085ccb0bcc44160a4232ca34e35331c366bc63a8c02b1fa" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/fbbc2f75-e230-4075-aab0-04e2451764e0.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220331%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220331T050142Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=a6c3cecbfb4a05e4f08575bbe4bf26c91052b382ea1dea41a1b5a64bc5fe5da9" ∫ 0 1 ( f ( x ) · g ( x ) ) d x ≠ ∫ 0 1 f ( x ) d x · ∫ 0 1 g ( x ) d x .

Let $f\left(x\right)=x,g\left(x\right)=x$.

Now,

$$\begin{gathered} {\int }_0^{1}f\left(x\right)dx={\int }_0^{1}x.dx \\ =\left[\frac{1}{2}-0\right] \\ =\frac{1}{2} \end{gathered}$$

And,

$$\begin{gathered} {\int }_0^{1}g\left(x\right)dx={\int }_0^{1}x.dx \\ ={\left[\frac{x^2}{2}\right]}_0^1 \\ =\frac{1}{2}\left[1-0\right] \\ =\frac{1}{2} \end{gathered}$$

So, 

$$\begin{gathered} {\int }_a^{b}f\left(x\right)dx·{\int }_a^{b}g\left(x\right)dx=\frac{1}{2}.\frac{1}{2} \\ =\frac{1}{4} \end{gathered}$$

Part c Step 2 . Hence,

$$\begin{gathered} {\int }_a^b\left(f\right(x)·g(x\left)\right)dx={\int }_0^{1}x.x.dx \\ ={\left[\frac{x^3}{3}\right]}_0^1 \\ =\left[\frac{1}{3}-0\right] \\ =\frac{1}{3} \end{gathered}$$

So, ${\int }_a^b\left(f\right(x)·g(x\left)\right)dx\ne {\int }_a^{b}f\left(x\right)dx·{\int }_a^{b}g\left(x\right)dx$

Therefore it is proved.