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Given is the graph of an exponential function $f\left(x\right) = \mathrm{exp}\left(-x\right)$. I need to find the area under it in an interval $A$. Is there a way to do this graphically without drawing many rectangles, like in the way you find the area under a linear function using just one rectangle.

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  • $\begingroup$ Almost certainly no. One of the many reasons integration and its associated Fundamental Theorem are so useful. $\endgroup$
    – Simon S
    Oct 19, 2015 at 17:53
  • $\begingroup$ Are we allowed to use the general results of integration theory in answering your question, if we do not do the particular integral? Note that most linear functions will require a trapezoid (which is half a rectangle), rather than a rectangle, to find the area under the graph. $\endgroup$ Oct 19, 2015 at 17:54
  • $\begingroup$ But is there a way that is faster (more accurate) than drawing many rectangles? Like drawing just one or two? $\endgroup$
    – Aaron Wild
    Oct 19, 2015 at 17:55
  • $\begingroup$ @RoryDaulton, Yes. $\endgroup$
    – Aaron Wild
    Oct 19, 2015 at 17:57

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In this diagram for $f(x)=e^{-x}$, $A$ has the coordinates $(a,0)$ and $B$ is the point $(b,0)$.

enter image description here

Point $A'$ is the point above $A$ on the graph of $f(x)$, $B'$ is above $B$ on the graph, $B''$ is on the horizontal line through $B'$ and the vertical line through $A$ and $A'$. Points $C$ and $D$ are to the left of points $A'$ and $B''$ so that $CB''$ and $DA'$ have lengths $1$.

Then rectangle $A'B''CD$ has the same area as that between the graph and the $x$-axis, and between points $A$ and $B$ (which I could not figure out how to shade in Geogebra).

The areas are equal due to the general integral equation

$$\int_a^b e^{-x}\,dx=e^{-a}-e^{-b}$$

Does this meet your needs?

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