Question
Consider the two functions f and g on [3, 9] such that J? f() dz = 13,J? g(.) dr = 7,J9 f(z) dz = 7 and J3 g(w) dx = 3.Evaluate J3 5f(c) dx:a. 6b 10CS 30d. 42e.None ofa-d
Consider the two functions f and g on [3, 9] such that J? f() dz = 13,J? g(.) dr = 7,J9 f(z) dz = 7 and J3 g(w) dx = 3. Evaluate J3 5f(c) dx: a. 6 b 10 CS 30 d. 42 e.None ofa-d
Answers
Let $\mathrm{f}(\mathrm{x})$ be differentiable in $\mathrm{R}$ and $\mathrm{f}(\mathrm{x})=2+\mathrm{x}^{2} \int_{0}^{2} \mathrm{f}(\mathrm{t}) \mathrm{d} \mathrm{t}+\int_{0}^{2} \mathrm{t} \mathrm{f}(\mathrm{t}) \mathrm{dt} .$ Then $\int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ is (a) $\frac{6}{19}$ (b) $\frac{3}{19}$ (c) $\frac{14}{19}$ (d) $\frac{-6}{19}$
Mhm. Mhm. So this problem here, they give us um an anti derivative of lower case, G is capital G of X plus C. And then they give us certain conditions, we know that G +04 is nine. GF six is 4 G of 96. They asked us to evaluate some definite integral. So the fundamental theorem of calculus, this is going to be G fx Evaluated from 6 to 4. So this is G of four -G six, which is nine minus four, that equals five. Second integral seven times the integral from 6 to 9 or the integral 69 of seven G of X. The easiest thing is just to write this. This is seven Times the integral from 6- nine. G F X dx. That's going to be seven G f x To evaluate it from 6- nine. So this is seven times GM nine -GF 6. It is So G of nine is 6 G of six, ah is four, so this is value of 14 And then the last one is the definite integral from 4- nine. Mhm. Of G of X plus three. So this is going to be if you split it into two intervals, the integral from 4 to 9 of G. Of X. Dx Plus. The integral from 4 to 9 of three d. x. So this is going to be G of nine -G. of four. Uh huh. Plus and that's just going to be three x Evaluated from 9- four. Yeah. And so if we look at our values G of nine, you have nine is six. So this is six minus G of four is 9, so 6 -9 plus three times 9 -4. So this is negative three Plus three times 5 15 -3. That answer is 12.
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Okay. What? We were told we have a function f and F is inoperable. And we're also told that the integral of 0 to 3 of F A Z D Z is equal to three, and the integral from 0 to 4 of F A C D. C is equal to seven. And based off of those two girls, we want to find the following. We want to find the integral from 3 to 4. Uh, empathy. Easy. Okay, so, um, we what we can do is subtract thes two into girls. And so we have the integral from 0 to 4 of f A Z. Easy, Morris. Subtract from it the integral from 0 to 3 of f of the easy. And so when I do from 04 and take away from 0 to 3, I'm left with 3 to 4. Because I have, um and so that's gonna be equal. Thio seven minus three, which is four. So we have, um now the integral from 3 to 4 of f A Z D Z is equal to four. And now what we want to find is the integral from 4 to 3 of empathy. Easy. Which is gonna be equal to the negative, Um, in a roll from 3 to 4 of f A C. D. C. Because, remember, any time I changed my upper low limits, I have to put a negative sign in front to this is gonna be equal to a negative four.
The execution we have to find the value of integration -3-3. F x dx. No we have given that G X is equal school Huaxi Square -8 X plus one. Now I'm going to find the value of g minus x. So I'm going to put here minus x. Of two into minus X. Or the minus eight into this is minus X plus one. So this will come out to be two X squared Plus air tax less one. Now effects will be equals to their taste. GX -1 gov minus X upon to not put the radio of G X. So G S is to access where minus air tax last one and G o minus X is equal to disease to exit square last air tax plus one upon too. So this will come out to be minus eight X. So this is our function F X. Now we can see that this is an old function. It is or function so Integration of -3-3, this is minus air text, the X. And this will be zero because the integration of all function is zero. So this is our answer for this, given to shin, and for this option is the correct answer. Thank you.