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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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Consider a real-valued function defined on a closed interval. One of the fundamental objectives in calculus is to determine the area under the graph of such a function. When an exact computation is not readily available, this area can be estimated by dividing the interval into a finite number of equal subintervals. Each subinterval corresponds to a rectangle whose width is the length of the subinterval and whose height is determined by the value of the function at a selected point within that...
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A car’s motion over time can be effectively analyzed using integral calculus, particularly through the concept of the definite integral applied to a velocity–time relationship. The definite integral describes how velocity accumulates over a specified time interval to produce total displacement. From a geometric perspective, this displacement is interpreted as the area under the velocity–time curve. Several key properties of definite integrals make it easier to analyze motion...
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Definite integrals are essential tools in calculus, used to quantify accumulated change over an interval. A common physical application is calculating the total displacement from a velocity-time graph. If a velocity function, v(t), describes the motion of an object over time, the definite integral gives the net displacement between times a and b. This integral corresponds to the signed area under the velocity curve between those two points.Two fundamental properties of definite integrals aid in...
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The highest and lowest values of a function, relative to a reference axis, are known as extreme values. These include absolute maximum and absolute minimum values, which represent the highest and lowest points the function reaches across its entire domain. Within a restricted portion of the function, the highest and lowest values are referred to as local maximum and local minimum values, respectively.Periodic functions, such as sine and cosine, show extreme values at infinitely many points due...
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The definite integral plays a critical role in understanding motion, particularly when calculating how far an object has traveled over time. Two important principles that emerge from this application are the Positivity Property and the Comparison Property of definite integrals. These properties provide intuitive physical interpretations based on velocity and displacement.Positivity Property of Definite IntegralsThe Positivity Property states that if an object’s velocity remains...
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Related Experiment Video

Updated: Feb 2, 2026

Establishment of a Primary Culture of Patient-derived Soft Tissue Sarcoma
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Local Control With Moderately Hypofractionated Definitive Radiotherapy Delivered With a Simultaneous Integrated Boost

J D Towler1, C David1, O Willis1

  • 1The London Sarcoma Service, University College Hospital, 235 Euston Road, London, NW1 2BU, UK.

Clinical Oncology (Royal College of Radiologists (Great Britain))
|January 31, 2026
PubMed
Summary
This summary is machine-generated.

Moderately hypofractionated definitive radiotherapy (MHDRT) offers an effective treatment option for inoperable non-extremity sarcomas. This approach shows promising local control rates with acceptable toxicity in patients unfit for or declining surgery.

Keywords:
ChordomaOsteosarcomaRadiotherapy dose fractionationSarcoma

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Area of Science:

  • Oncology
  • Radiation Oncology
  • Surgical Oncology

Background:

  • Surgery is the primary treatment for soft tissue sarcomas (STS) and bone sarcomas.
  • A significant number of patients with non-extremity tumors are not candidates for resection.
  • Limited outcome data exist for definitive radiotherapy in these cases, especially with hypofractionation near organs at risk.

Purpose of the Study:

  • To evaluate the institutional experience with moderately hypofractionated definitive radiotherapy (MHDRT).
  • To assess the efficacy and toxicity of MHDRT in patients with inoperable non-extremity STS and bone sarcomas.
  • To investigate MHDRT in patients unfit for or declining surgery.

Main Methods:

  • Fifty-nine adult sarcoma patients received MHDRT between July 2021 and September 2024.
  • Treatments involved 60 courses using photon or proton therapy over 28 fractions.
  • Dose levels included 50.4 Gy and a simultaneous integrated boost (SIB) of 63 Gy for STS or 70 Gy for bone sarcomas.

Main Results:

  • At 1-year follow-up (median 17.7 months), local control was 90.8% for STS, 100% for chordoma, and 55.6% for high-grade bone sarcomas.
  • Acute and late grade 3 toxicities were observed in 8.3% and 10% of patients, respectively.
  • Encouraging early local control rates were achieved across tumor types.

Conclusions:

  • MHDRT is a viable radical treatment option for inoperable non-extremity sarcomas.
  • The treatment demonstrates acceptable toxicity profiles in this patient cohort.
  • MHDRT provides encouraging early local control, addressing a critical need for alternative therapies.