- If \(\ell'(t)\) represents the instantaneous growth rate of the length of a worm at a time \(t\) months into its life, what does the integral \(\int_3^5 \ell'(t) \,\mathrm{d}t\) represent?
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Evaluate these indefinite integrals.
Don’t forget your \(+C\)s!
\(\displaystyle\int t^4+t^3+t^2+t+1 \,\mathrm{d}t\)\(\displaystyle\int 7\sin(x) \,\mathrm{d}x \)\(\displaystyle\int 3\csc^2(t) \,\mathrm{d}t \)\(\displaystyle\int 4\sec(x)\tan(x) \,\mathrm{d}x\)\(\displaystyle\int \frac{\mu-\mu^2}{\mu-1} \,\mathrm{d}\mu \)\(\displaystyle\int (1-x)^3 \,\mathrm{d}x\)\(\displaystyle\int 2\sqrt[3]{x} + \frac{3}{x^6} \,\mathrm{d}x\)\(\displaystyle\int \frac{1}{\sqrt[3]{x}} \,\mathrm{d}x \)\(\displaystyle\int \sqrt{x-7} \,\mathrm{d}x \)\(\displaystyle\int \sqrt{2x-7} \,\mathrm{d}x \)
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What are the exact values
of the following definite integrals?
\(\displaystyle\int\limits_{1}^{2} t^4+t^3+t^2+t+1 \,\mathrm{d}t\)\(\displaystyle\int\limits_{\pi/6}^{\pi/4} 4\sec(x)\tan(x) \,\mathrm{d}x\)\(\displaystyle\int\limits_{27}^{125} \frac{1}{\sqrt[3]{x}} \,\mathrm{d}x \)
-
Suppose I’m standing on the side of a Nevada freeway
with a radar gun in hand.
A Maserati is cruising towards me.
I point my radar gun towards the car to measure its speed and,
noticing this, the driver of the Maserati floors it.
The car speeds past me and disappears into the distance,
the radar gun losing track of the car exactly twelve seconds
after I first pointed it towards them.
My radar gun, being quite technologically advanced,
used the data it collected on the speed of the Maserati
and worked out a model \(f(t)\)
for the car’s speed \(v\) (measured in ft/s)
at any time \(t\) during those twelves seconds:
\[ v \;=\; f(t) = 120 + 2t^2 - \frac{1}{12}t^3\,. \]
- What’s the fastest the Maserati was going in those twelve seconds?
- What was the Maserati’s average acceleration over those twelve seconds?
- How far did the Maserati travel in those twelve seconds?
- Write down a formula for the function \(F(t)\) that expresses the distance the Maserati is from Reno, assuming the highway and the Maserati are headed in a straight line away from Reno, and that the Maserati is already \(96024\) feet from Reno at time \(t = 0.\)
-
Recall that an even function \(f\)
has the property that \(f(x) = f(-x).\)
Supposing that \(f\) is an even function
and \(\int_0^2 f(x)\,\mathrm{d}x = 7,\)
what must the value of the following definite integrals be?
\(\displaystyle\int\limits_{0}^{2} -3f(x) \,\mathrm{d}x \)\(\displaystyle\int\limits_{2}^{0} f(x) \,\mathrm{d}x \)\(\displaystyle\int\limits_{0}^{2} 2x+f(x) \,\mathrm{d}x \)\(\displaystyle\int\limits_{-2}^{2} f(x) \,\mathrm{d}x \)
-
What are the exact values
of the following definite integrals?
\(\displaystyle\int\limits_{-1}^{5} |x-1|-1 \,\mathrm{d}x \)\(\displaystyle\int\limits_{-100}^{100} \arctan(x) \,\mathrm{d}x \)\(\displaystyle\int\limits_{-2}^2 \sqrt{4-x^2} \,\mathrm{d}x \)
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Recall the first Fundamental Theorem of Calculus,
and write down formulas for the derivatives
of the following functions.
\(\displaystyle a(x) = \int\limits_{-\pi}^{x} t^3-t \,\mathrm{d}t \)\(\displaystyle b(x) = \int\limits_{x}^{7} t^2\sec(t) \,\mathrm{d}t \)\(\displaystyle c(x) = \int\limits_{0}^{\tan(x)} \sin(t) \,\mathrm{d}t \)
- Find a geometry-based approximation of the value of \(\int_{-6}^{6} \sqrt{x^2+1} \,\mathrm{d}x \) that accurate to within \(10\%\) of its true value of about \(27.8075359.\)
- If \(g''(x) = \sqrt{x},\) \(g(1) = \frac{1}{2},\) and \(g(0) = -16,\) what must a formula for \(g(x)\) be?
- Sketch a graph of the antiderivative \(F\) of the function \(f(x) = \sec(x)\tan(x)\) defined for all real numbers except the numbers of the form \(\left(\frac{1}{2} + k\right)\pi\) for an integer \(k\) such that \(F(n\pi) = -n\) for every integer \(n.\)
Challenges
-
James Stewart
For real numbers \(a\) and \(b\) such that \(0 \lt a \lt b\), calculate an expression for \(\int_a^b \lfloor x \rfloor\,\mathrm{d}x\) in terms of \(a\) and \(b\). -
James Stewart
Prove that \[ \frac{1}{17} \;\;\leq\;\; \int\limits_1^2 \frac{1}{1+x^4} \,\mathrm{d}x \;\;\leq\;\; \frac{7}{24}\,. \]