WEBVTT
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right. This problem. We need to find area
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of the shaded regions. So buy formula. We
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know. Ah, if we want to inter grow
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instead ofthe X first of the red on the integral
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. So the area Hey, no e coast too
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into grow in terms ofthe X And we know the
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shaded area X goes from zero to one so we
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can write the bound it pounds from zero to one
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. Okay. Ah, for here. Ah,
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the formula is upper curve minus the lower occur.
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So this will be you two The power X linus
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x times e to the power X square. So
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our next step basically is Try toe Aah! Evaluate
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this, integral. All right, So we can
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do it. Except to separately the first term.
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Well, the ranger grow from zero to one into
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the power ext the ex minus integral from zero to
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one times to the power X square, the ex
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. Okay. And the first that we know the
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anti directly about that. This is e to the
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Power X. So this is such a yeah to
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the power X evaluate acts because one minus X equals
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zero. Okay. And this term here we can
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bring this acts to the differential that will give us
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X square. So before we do that, we
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need two times two. So what we trick they're
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using here every times two and it divided by two
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. So the second term will be behalf ties into
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grove the roads one you two the power X square
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King X square. And as we can see this
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if we use change of arable say this explorer to
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be another variable t Then this is you to the
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party. So the auntie do it Who will be
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itself. So this term Ah, you free right
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here. No B negative One half times into the
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power X square evaluated and ex ecos one minus x
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Go to zero. Right? So, in other
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words, this is, um So next page the
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first term, remember, is need to the power
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axe. You're only right from one minus X equals
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in Cyril zero um, minus one half he to
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the Power X square. Valerie had taxi. Could
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one minus X equals zero. So no and serve
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. It will be the first term. Here is
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I e to the power of one. Linus Ito
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, the power zero which is what and the second
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term and second term is minus one half times the
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first of eyes that you two, the power one
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. So it's just e to the tower one minus
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into a powered zero where she's far and as you
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can see, answer will be one half times minus
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one, right.