Evaluate definite integrals and calculate areas bounded by curves or axes, splitting intervals where signs require it.
Tę umiejętność sprawdza 35 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Rectangle k has width 1, spans k − 1 ≤ x ≤ k, and has height (2k + 1)2. A quadratic curve y=f(x) passes through their top-right corners. (i) State f(x). (ii) Compare areas to show . (iii) Use a similar area comparison to show…
Find the area bounded by y=f(x), the axes and x=3π.
Find the area of R.
If V = 135π2 cm3, find d exactly.
The positive intersection x-coordinates are a<b. Find the area enclosed by the curves between a and b.
Show that the area bounded by y=f(x), the x-axis and x=0, x=1 is ln(486/125).
(i) Use 11c to approximate . (ii) Hence state the area between the polynomial y = −7/4 + (17/16)x − (37/64)x2, the x-axis and x=0, x=1.
The tangent referred to here is the tangent to y=f(x) at x=2. This tangent meets the curve again at a point P distinct from the tangency point. Find (i) P; (ii) the enclosed area.
A region is bounded above by y = 1 + e2x, below by y = 1 − 2e2x, and by the vertical lines x = 0 and x = ln 2. Find its exact area.
At the beginning of 2021, a sealed reservoir holds 420000 litres. Its modelled rate of water loss is described by litres per year, where t is years after the beginning of 2021. Find the volume predicted for the beginning of…
A separate function is h(x) = 9x2/(3x + 2), x ≠ −2/3. Region R is bounded by its curve, the line y = 1, and x = 1, x = 2. (i) Establish . (ii) Hence or otherwise, calculate the area of R in the form p + qln r, with positive…
The cart modelled by u finishes first. How far is the other cart from its endpoint at that instant?
Find the area of the shaded region bounded by portions of y = g(x), y = g−1(x) and the positive coordinate axes.
Find the area of the enclosed region.
Take k=6. Find the area enclosed by the graph of f, its normal at x=−4, and the y-axis, on the interval −4≤x≤0.
Find the area enclosed by the graphs.
Find the exact area of R.
State (i) ; (ii) .
It is given that A_2=1 and A_3=3. Use calculator integration with a suitable finite upper bound to estimate (i) A_4; (ii) A_5.
Find the distance travelled between t1 and t2.
Determine whether overflow occurs before the first flush. Justify your answer.
Find the area of R_n exactly as a function of n.
Hence show that . You may use the primitive x3(ln x)2/3 − 2x3ln x/9 + 2x3/27.
Hence show that . You may use the primitive x3(ln x)2/3 − 2x3ln x/9 + 2x3/27.
Use the first two nonzero terms F(x) ≈ (e + 1) − 8(e + 1)x2 to show that .
Use the first two nonzero terms F(x) ≈ (e + 1) − 8(e + 1)x2 to show that .
Evaluate .
Find its total distance travelled over the given interval.
A continuous random variable X has density for a≤x≤c, for c<x≤b, and 0 elsewhere. Assume a<c<b and c ≥ (2b + a)/3. Find its median in terms of a,b,c.
Hence approximate using terms up to degree 3 in the Maclaurin expansion of f. For standalone use, f(u) = u + 2u2 + 11u3/6 + ⋯.
Find the total area between the graph and the x-axis from x=0 to x=2.
For m=3/2,c=2,h=4, show that .
In the general frustum, the smaller radius is r=c. (i) Give the larger radius R. (ii) Express the slant height l in terms of m,h. (iii) Use the supplied integral to prove A = π(r + R)l.
The section for −r/2 ≤ x ≤ r/2 is rotated about the x-axis. Using the supplied integral, show that the curved surface of this spherical belt has area 2πr². Do not include the flat end disks.
Let for −r≤x≤r, with r>0, and let k>0. (i) Describe the transformation from y=f(x) to y=kf(x). (ii) State both x-intercepts of y=kf(x). (iii) Find its derivative for −r<x<r. (iv) The full transformed arc is rotated about the x-axis.…
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