Revise Introduction to integration and area under a curve for IB Mathematics: Applications and Interpretation (5.3)

Differentiation turns $x^3$ into $3x^2$. Integration undoes this: starting from $3x^2$ it should recover $x^3$. The catch is that $x^3+1$, $x^3-7$ and $x^3$ all have the same derivative $3x^2$, because the derivative of a constant is zero. Reversing a derivative therefore cannot tell you which constant was originally present, so the result is a whole family of functions written with a constant of integration $+C$. This reversed process, with no limits attached, is called the indefinite integral, written $\int f(x)\,dx$.

$\displaystyle\int f(x)\,dx = F(x) + C \quad\text{where}\quad F'(x) = f(x)$

You can always check an integral by differentiating the answer: if you get the integrand back, you were right.

Reversing the rule 'multiply by the power, then reduce the power by one' gives the rule for integrating a power of $x$: add one to the power, then divide by the new power.

$\displaystyle\int ax^n\,dx = \frac{a x^{\,n+1}}{n+1} + C, \qquad n \neq -1$

Integration on its own leaves the unknown constant $+C$. If a question gives you one extra fact — a point the curve passes through, or an initial value — you can pin the constant down and recover the exact function. The routine is always the same: integrate to get the general form, substitute the known values, solve for $C$, then write the specific function.

Attach a lower limit $a$ and an upper limit $b$ to the integral and it becomes a definite integral, $\int_a^b f(x)\,dx$, which evaluates to a single number rather than a family of functions. Its meaning is geometric: for a curve lying above the x-axis, it equals the area of the region bounded by the curve $y=f(x)$, the x-axis, and the vertical lines $x=a$ and $x=b$. You can picture that area as a stack of infinitely thin rectangles of height $f(x)$ and tiny width, summed across the interval.

$\displaystyle\text{Area} = \int_a^b f(x)\,dx = \big[F(x)\big]_a^b = F(b) - F(a)$

IB awards marks analytically: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if the arithmetic later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct final step carried out on your own earlier (wrong) value still scores, and equivalent correct forms — including a value read straight from the GDC — are accepted. All of that protection exists only if your method is on the page. Study how each mark below is earned by a specific line.

Everything here is one idea seen from two sides: integration reverses differentiation, and evaluated between limits it measures area. That single tool carries straight into applications — integrating a velocity to recover displacement (remembering $+C$ and an initial condition), integrating a rate to find a total accumulation, and finding the area between two curves by integrating the top curve minus the bottom. Master the habit — reverse the power rule, fix the constant from a condition, set up the definite integral with the right limits, then evaluate by hand or on the GDC — and the rest of integral calculus becomes variations on a method you already own.