Integration (HSC SSCE Mathematics Advanced): Quizzes

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Challenge – Proving the Fundamental Theorem
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What two fundamental operations of calculus does the fundamental theorem connect?

Differentiation and integration

What is the value of the signed area function A(a)=aaf(t)dtA(a) = \int_a^a f(t) \, dt?

00

In signed area, how do areas below the xx-axis contribute to the integral?

Negatively

What does ddxaxf(t)dt\frac{d}{dx} \int_a^x f(t) \, dt equal according to the differential form?

f(x)f(x)

What is ddx0x(t2+1)3dt\frac{d}{dx} \int_0^x (t^2 + 1)^3 \, dt using the differential form?

(x2+1)3(x^2 + 1)^3

What formula does the integral form provide for abf(x)dx\int_a^b f(x) \, dx?

F(b)F(a)F(b) - F(a)

What technique does the proof of the differential form use?

Sandwiching with rectangles

How do any two primitives of the same function differ?

Only by a constant

What property must f(x)f(x) have for the fundamental theorem to apply?

Continuous in the interval

The signed area function A(x)=axf(t)dtA(x) = \int_a^x f(t) \, dt is what type of function?

An antiderivative of f(x)f(x)

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