[Math] Finding the lower and upper Riemann Sums

definite integralsriemann sum

I'm having a hard time figuring this questions out. I've looked on google and on the book and so far I haven't gotten a good explanation for this questions. I know they aren't hard and are probably easy but I've haven't gotten a good explanation.

  1. Find the lower and Upper Riemann Sums $L(P)$ and $U(P)$ for the function $f(x)=x^2$ on the interval $[0,1]$ using the partition $P=\{0,\frac12,
    \frac34,1\}$

  2. Given that $\int_1^3 f(x)~dx=4$ and $\int_1^5 f(x)~dx=7$,
    find $\int_3^5 f(x)~dx$.

  3. Given that $\int_1^3 f(x)~dx=4$ and $\int_1^3 g(x)~dx=2$,
    find $\int_1^3 3f(x)-g(x)~dx$.

Best Answer

Just so this isn't hanging around unanswered forever.

  1. Choose the minimum function value over each interval for $L(P)$, and the maximum for $U(P)$: $L(P)=0^2(\frac12)+(\frac12)^2(\frac14)+(\frac34)^2(\frac14) = \frac{13}{64}. U(P)=(\frac12)^2(\frac12)+(\frac34)^2(\frac14)+1^2(\frac14)=\frac{33}{64}$

  2. $\int_3^5 f(x)~dx = \int_1^5f(x)~dx - \int_1^3 f(x)~dx = 7 - 4 = 3$

  3. $\int_1^3 3f(x)-g(x)~dx =\int_1^3 3f(x)-\int_1^3g(x)~dx =3\int_1^3 f(x)-\int_1^3g(x)~dx = 3(4)-2=10$.

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