⧼exchistory⧽
15 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
If the function [math]f[/math] is continuous at every point of the interval [math][a,b][/math] and may cross the [math]x[/math]-axis at a finite number of points in the interval. Let [math]P^+[/math] and [math]P^-[/math] have their usual meaning [as in formula].
- Is [math]|f(x)|[/math] continuous at every point of [math][a,b][/math]?
- Show that
[[math]] \mbox{''area''}(P^+ \cup P^-) = \int_a^b |f(x)| \; dx . [[/math]]
BBy Bot
Nov 03'24
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[/math]
Find the area of the region bounded by the parabola [math]y = x^2[/math], the [math]x[/math]-axis, and the line tangent to the parabola at the point [math](2, 4)[/math]. Do the problem
- using [math]x[/math] as the variable of integration.
- using [math]y[/math] as the variable of integration.
BBy Bot
Nov 03'24
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[/math]
Do Problem Exercise for the line tangent to the parabola at the general point [math](a, a^2)[/math].
BBy Bot
Nov 03'24
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[/math]
Express the area of the ellipse [math]\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1[/math] as a definite integral of a function of [math]x[/math], and as a definite integral of a function of [math]y[/math]. (The resulting indefinite integrals cannot be evaluated with the theory so far developed.)
BBy Bot
Nov 03'24
[math]
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[/math]
Find the area of the shaded region in Figure. The curves are parabolas. The inscribed square has area [math]4[/math], and the circumscribed square has area [math]16[/math].