anonymous
  • anonymous
Can you explain definite integrals?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
Integrals are often used to determine the area under a curve/surface depending on the number of integrals invovled. Assuming that just one integral is being used, then \[\int\limits_{ }^{ }f(x)dx\] is the area between the curve f(x) and the x axis. When we have a definite integral, values are given which allow the area under the curve to be calculated between two specific points: \[\int\limits_{a }^{ b}f(x)dx\] This gives the total area under the curve starting at x = a and ending at x=b.
amistre64
  • amistre64
id agree in principal, but the first part is alittle askew
amistre64
  • amistre64
the first part provides us with a family of related functions that can be used in a variety of ways

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amistre64
  • amistre64
\[\int f(x)dx;\ knowing\ that\ F(0)=5\]
anonymous
  • anonymous
How do I solve this problem?? \[\int\limits_{-1}^{5}(1+3x)dx\]
amistre64
  • amistre64
integrate it as usual; to find the desired function; then evaluate that function using the limits as: F(5)-F(-1)
anonymous
  • anonymous
How do I integrate it?
amistre64
  • amistre64
do you know how to find a derivative?
anonymous
  • anonymous
yes
amistre64
  • amistre64
integration is just undoing a derivative; also known in some earlier circles as antidifferentation
amistre64
  • amistre64
\[\frac{d}{dx}? = Constant\] \[? =\int Constant\]
amistre64
  • amistre64
\[\frac{d}{dx}\frac{x^{n+1}}{n+1}=x^{n}\] \[\frac{x^{n+1}}{n+1}=\int x^{n}\]
anonymous
  • anonymous
so I'm supposed to do that with 1+3x? like the integral would be x+\[3/2x ^{^{2}}\]
amistre64
  • amistre64
exactly
amistre64
  • amistre64
\[F(x)=x+\frac{3}{2}x^2\] to find the definite part of it; use the limits F(2)- F(-1)
amistre64
  • amistre64
\[F(2)=2+\frac{3}{2}(2)^2\]\[-F(-1)=-(-1+\frac{3}{2}(-1)^2)\]
anonymous
  • anonymous
so now I have 5.5. Is that right? and if so, what do I do next?
amistre64
  • amistre64
2+6+1-3/2 = 9 - 3/2 = 8 - 1/2 = 7.5
amistre64
  • amistre64
what you do next is put 7.5 in the answer box and move to the next problem ;)
anonymous
  • anonymous
How did you get 7.5? and the back of the book says that the answer is 42...
amistre64
  • amistre64
i plugged in the limits, and did the math; as typed out above
amistre64
  • amistre64
and 42 is not the answer unless there is more to the problem then you have posted
amistre64
  • amistre64
i see a mistake i did; is read 5 as a 2
anonymous
  • anonymous
So how do I fix that? What do I do differently?
anonymous
  • anonymous
I put in 5 instead of 2, and I got 43, not 42...
amistre64
  • amistre64
\[F(5)-F(-1)=5+\frac{3}{2}(5)^2-(-1+\frac{3}{2}(-1)^2)\]\[\hspace{10em} =5+\frac{75}{2}+1-\frac{3}{2}\]
amistre64
  • amistre64
6 + 72/2 = 6 + 36 = 42
anonymous
  • anonymous
ooh yay it's right!! Thanks so much!!
amistre64
  • amistre64
yep, yw ;)
anonymous
  • anonymous
How can I solve the problem using the following formula? \[\int\limits_{a}^{b}f(x)dx=\lim_{n \rightarrow \infty} \sum_{i=1}^{n}f(x _{i})\Delta x\]

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